KYEI, YAW. Higher-Order Cartesian Grid Based Finite Difference Methods for Elliptic Equations on Irregular Domains and Interface Problems and their Applica-tions. (Under the direction of Kazufumi Ito.)
This thesis describes higher-order finite difference methods for solving elliptic equations on irregular domains with general boundary conditions and the correspond-ing elliptic interface problems. We develop second and fourth order methods for two and three dimensions using uniform Cartesian grids. However, with an irregular do-main we cannot apply the standard finite difference schemes directly at the grid points near the boundary and therefore some treatment is required in order to use the uni-form Cartesian grids. Our approach involves modifying the standard finite difference schemes. In particular, we use the standard five-point and standard compact nine-point stencil schemes for the second and fourth order methods, respectively. That is, on the standard stencils that contain the boundary, we carry out the modification by applying the continuation of solution from the inside of domain to the outside.
The method of continuation of solution uses Taylor series expansion of the solu-tion about selected boundary points, the equasolu-tion and the boundary values of the local and their nearby boundary points. Naturally, second and fourth order Taylor series expansions about the boundary points are used for the second and fourth order methods respectively.
Our methods have a unified and an effective approach to deal with general bound-ary conditions and capture the boundbound-ary and its local geometrical properties by the level set function and the local coordinate system at the boundary points. The re-sulting finite difference system matrices of our methods remain symmetric positive definite and maintain the sparsity of the standard finite difference schemes.
demon-As one of the essential applications of our method, we design a state feedback controller for the Dirichlet boundary control problem of the heat equation. By the standard LQR theory of the optimal state-feedback design, we solve the associated Riccati equation where the ODE system resulting from our fourth order method of semi-discretization serves as the state equation. Conventional second order methods have system matrices of higher dimensions which makes the Riccatti equation almost impractical to solve numerically. But with our nine-point compact fourth order meth-ods, we capture the essential properties of the equation through our low order system matrices and demonstrate the capability of our approach through our computations.
IRREGULAR DOMAINS AND INTERFACE PROBLEMS AND
THEIR APPLICATIONS
by Yaw Kyei
a dissertion submitted to the graduate faculty of north carolina state university
in partial fulfillment of the requirements for the degree of
doctor of philosophy
department of mathematics
raleigh April 28, 2004
approved by:
chair, Dr. Kazufumi Ito Member, D. Zhilin Li
Yaw Kyei was born and raised in Asuafu Ashanti, a small town in Ghana. He had his elementary school education locally and completed O’level secondary education at Nsutaman Catholic secondary school, Nsuta Ashanti in 1988. He completed A’level secondary education at Kumasi Academy, Kumasi in 1990. He entered university of Ghana in 1992 and graduated with a Bachelor of Science honours degree in Mathe-matics and Statistics at the top of his class in 1996. He then served as a teaching assistant for one year at the mathematics Department, University of Ghana during the 1996/97 academic year teaching freshman mathematics to medical and social science students. During the 1997/98 academic year, Yaw was awarded a maiden opportunity of an exchange program to North Carolina State university for a possi-bility of entering into a mathematics graduate program. He joined the mathematics department in 1998, earned a Master of Science degree in Applied Mathematics in May 2000 and a Doctor of Philosophy degree in Computational Applied Mathematics in April 2004 from North Carolina State University in Raleigh, North Carolina.
I am very grateful to my advisor Dr. Kazufumi Ito for his many invaluable assistance, guidance, encouragement, coaching and constant support throughout my Ph.D study and research. It is a rewarding and an enjoyable experience working with him. I very much appreciate the friendship and the relationship that we have cultivated and de-veloped through many research and life discussions which leaves a lasting impression. I would also like to express my appreciation and gratitude to my advisory com-mittee members being Dr. Zhilin Li, Dr. Hien T. Tran and Dr. Ralph Smith for their directions, comments and suggestions.
I would like to sincerely thank Dr. Ernest L. Stitzinger, the Mathematics Gradu-ate Programs Administrator for his assistance throughout my study at NCSU. Sincere thanks are also extended to Ms. Denise Seabrook, the Mathematics Graduate Pro-grams Secretary for her generous help during my study at NCSU. I am also thankful to the faculty, staff, and fellow students in the Mathematics Department for providing me with such an excellent education.
My sincere thanks go to my family and friends in Raleigh and back home in Ghana, especially my parents Kwabena Dapaah and Martha Amankwah for their longtime confidence and support. Finally and more importantly, I deeply appreciate my wife, Charlotte Kyei, for her constant love, understanding and support.
List of Tables vii
List of Figures ix
1 Introduction 1
1.1 The Model Problem . . . 4
1.1.1 Physical Background of Diffusion Processes . . . 5
1.2 Overview of Applications considered . . . 7
1.2.1 Time Dependent Diffusion Problems . . . 8
1.2.2 The Eigenvalue Problem . . . 8
1.2.3 Application to Boundary Control . . . 10
1.2.4 Extension to Elliptic Interface Problems . . . 10
1.3 Other Related Work . . . 11
1.4 Outline of Thesis . . . 13
2 The Second Order Method 15 2.1 Second Order method in Two Dimensions . . . 17
2.1.1 Dirichlet Boundary Conditions . . . 21
2.1.2 Neumann Boundary Conditions . . . 26
2.1.3 Robin Boundary Conditions . . . 30
2.1.4 Addition of a Potential term . . . 32
2.2 Second Order Method in Three dimensions . . . 32
2.2.1 The Choice of Local Coordinate Directions . . . 33
2.2.2 Dirichlet Boundary Conditions . . . 34
2.2.3 Neumann Boundary Conditions . . . 35
2.3 Implementation of the Second Order Method . . . 38
2.4 Numerical Results for Two Dimensions . . . 41
2.4.1 Order Calculations of the Numerical Schemes . . . 42
2.4.2 Results of Numerical Tests . . . 42
3 The Fourth Order Method 51 3.1 A Derivation of the Compact Nine-Point Stencil Fourth Order Finite
Difference Scheme . . . 51
3.1.1 Error Analysis . . . 56
3.2 The Fourth Order Method in Two Dimensions . . . 59
3.2.1 Dirichlet Boundary Conditions . . . 62
3.2.2 Neumann Boundary Conditions . . . 66
3.3 Implementation of the Fourth Order Method . . . 70
3.3.1 Numerical Results . . . 74
3.4 Summary . . . 79
4 Applications to the Heat equation and Eigenvalue problems 80 4.1 Semi-Discretization of the Linear Diffusion Equation . . . 81
4.1.1 Homogeneous in Space and Time Boundary Value . . . 83
4.1.2 Homogeneous in Space Boundary Value . . . 88
4.1.3 General Boundary Value . . . 93
4.2 Application to Eigenvalue Problems . . . 98
4.2.1 Eigenvalues and Eigenfunctions of a Circular Domain and the Bessel’s functions . . . 98
4.2.2 Eigenvalues and Eigenfunctions of other Irregular-Shaped Discs 101 4.2.3 Summary . . . 108
5 Application to Boundary Control Problems 111 5.1 The Control Problem Description . . . 112
5.2 Theoretical Justification of The Method . . . 114
5.3 The State Feedback Control . . . 118
5.4 Numerical Results . . . 119
5.5 Summary . . . 125
6 Extension to Variable Elliptic Equations and Interface Problems 126 6.1 The Variable Coefficient Elliptic Equation . . . 130
6.2 Domain with an Interface . . . 131
6.3 Numerical Experiments . . . 135
6.4 Summary . . . 139
7 Conclusions 140
2.1 Second Order Error Analysis for Dirichlet Conditions & Boundary Ge-ometry 2.4(a) with Example 2.4.2 . . . 44 2.2 Second Order Error Analysis for Dirichlet Conditions & Boundary
Ge-ometry 2.4(b) with Example 2.4.2 . . . 45 2.3 Second Order Error Analysis for Newmann Conditions & Boundary
Geometry 2.4(b) with Example 2.4.1 . . . 45 2.4 Second Order Error Analysis for Newmann Conditions & Boundary
Geometry 2.4(b) with Example 2.4.2 . . . 46 2.5 Second Order Error Analysis for Newmann Conditions & Boundary
Geometry 2.4(c) with Example 2.4.2 . . . 46 3.1 Fourth Order Error Analysis for both Variants with Dirichlet Boundary
conditions & Boundary Geometry 2.4(a) with Example 2.4.1 . . . 75 3.2 Fourth Order Error Analysis for both Variants with Dirichlet Boundary
conditions & Boundary Geometry 2.4(b) with Example 2.4.1 . . . 76 3.3 Fourth Order Error Analysis for both Variants with Dirichlet Boundary
Conditions & Geometry 2.4(b) with Example 2.4.2 . . . 76 3.4 Fourth Order Error Analysis for Variant One with Newmann Boundary
conditions on Geometry 2.4(b) for Examples 2.4.1 and 2.4.2 . . . 77 3.5 Fourth Order Error Analysis for Variant One with Newmann
Condi-tions on Geometry 2.4(a) for Examples 2.4.1 and 2.4.2 . . . 77 4.1 First 20 Eigenvalues against Estimates from the Fourth Order Method
for a Circular Domain . . . 100
4.3 Estimates of the first 20 Eigenvalues of 2DIrregular domains from the Fourth Order Method . . . 107 4.4 First two Eigenvalues of five Square Domains with Hollow Spaces . . 108 4.5 Gap Between first two Eigenvalues of the Square Domain with Hollow
Spaces (Fig 4.2) . . . 109 5.1 Closed Loop Eigenvalues of the Circular Domain with one Rectangular
RegionΩ of Regulation . . . 121b 5.2 Closed Loop Eigenvalues for Square Domain with Circular Hollow
Space . . . 122 5.3 Closed Loop Eigenvalues for Square Looking Domain with Regulation
at Center . . . 124 6.1 Error Analysis of the Fourth Order M-matrix 9-point Compact Scheme
for the variable Elliptic equation with Dirichlet conditions on a rect-angular domain . . . 136 6.2 Error Analysis of the Fourth Order M-matrix 9-point Compact Scheme
for the interface problem with jump sizes |β+ −β−| = 3,8,14 and
interface geometry Figure 2.4(b) . . . 137 6.3 Error Analysis for the Fourth Order M-matrix 9-point Compact Scheme
for the interface problem with jump sizes |β+−β−|= 29,99,999 and
2.1 A portion of the boundary intersecting 5−point standard stencil. . . 18
2.2 An irregular grid point with its projected point and the local coordinate transformation. . . 19
2.3 Local boundary representation of Γ at the projected point. . . 20
2.4 Irregular Boundary Geometries . . . 43
2.5 Least Squares Estimates of Orders for Dirichlet Boundary conditions on Geometry 2.4(a) corresponding to Table 2.1 . . . 44
2.6 Least Squares Estimates of Orders for Dirichlet Boundary conditions on Geometry 2.4(b) corresponding to Table 2.2 . . . 45
2.7 Least Squares Estimates of Orders for Neumann Boundary conditions on Geometry 2.4(b) corresponding to Table 2.3 . . . 46
2.8 Least Squares Estimates of Orders for Neumann Boundary conditions on Geometry 2.4(c) corresponding to Table 2.5 . . . 47
3.1 The square domain for the Compact 9−point stencil. . . 52
3.2 The Compact nine-point stencil. . . 53
3.3 Discretized Weights on the Compact 9-point stencil . . . 57
3.4 A portion of boundary intersecting the Standard Compact 9-point sten-cil . . . 60 3.5 Least Squares Estimates of Orders for both Variants with Dirichlet
Boundary conditions on Geometry 2.4(a) corresponding to Table 3.1 75 3.6 Least Squares Estimates of Orders for both Variants with Dirichlet
Boundary conditions on Geometry 2.4(b) corresponding to Table 3.2 76
corresponding to Table 3.4 . . . 77 3.8 Least Squares Estimates of Orders using Variant One for Examples
2.4.1 and 2.4.2 with Neumann Boundary conditions on Geometry 2.4(a) corresponding to Table 3.5 . . . 78 4.1 Comparison of the Estimates from the Fourth Order Method against
Exact Eigenvalues for n= 11 through n= 81. . . 99 4.2 Eigen-Properties of a Square Domain with 2 Hollow spaces . . . 102 4.3 Eigen-Properties of a Butterfly Looking Domain . . . 103 4.4 Eigen-Properties of a Square Domain with a Circular Hollow Space . 104 4.5 Eigen-Properties of a Square Domain with a Hollow Square-Shaped
Curved Boundary Space . . . 105 4.6 Eigen-Properties of a Square-Shaped Curved Boundary Domain . . . 106 4.7 Square Domains with Five Hollow Spaces of Decreasing Sizes . . . . 109 5.1 Circular Domain with Regulation on a Square at the Center . . . 120 5.2 Square Domains with Circular Hollow Space and the four Corners for
Regulation . . . 122 5.3 Square Looking Curved Boundary Domain with Regulation at Center 123 6.1 A portion of Domain Containing the Interface and Showing the Local
Coordinate Directions . . . 129 6.2 Least Squares Estimates of Orders for interface problem with interface
Geometry of Figure 2.4(b) and Dirichlet Boundary conditions on the Rectangular boundary outside . . . 137 6.3 Computed solution for the interface problem with the diamond-shaped
interface Geometry with β+ = 4, β−= 1 . . . 138
Introduction
Many problems in applications of physics and mathematics involve partial differential equations with initial and boundary conditions. Of core importance to us in this thesis, are the boundary value problems of the elliptic type. We investigate the solutions of these types of problems on domains that are in general non rectangular. Specifically in this work, we design and develop uniform Cartesian grid based finite difference methods for solving partial differential equations of the elliptic type on irregular domains with general boundary conditions. Our aim is to obtain higher order methods which are efficient, flexible and easier to implement.
In particular, we develop second and fourth order accurate finite difference meth-ods for solving elliptic equations on irregular domains Ω. Our methmeth-ods handle gen-eral boundary conditions in a systematic manner in two as well as three dimensions. We then apply our fourth order methods to semi-discretize the time-dependent heat equations on non rectangular domains. To validate the discretization of the parabolic equations and our numerical methods, we use the fourth order method to solve the associated eigenvalue problem to demonstrate the efficiency and capabilities of the methods. Subsequently, we use the fourth order methods to design optimal state feedback controller for the Dirichlet boundary control problem of the time-dependent heat equation on the non rectangular boundary of the domains.
We now briefly describe the outline of our proposed method. To develop our second order method in two dimensions, we use the second order central difference scheme
4ui,j−ui+1,j−ui−1,j−ui,j+1−ui,j−1
h2 =fi,j (1.0.1)
for discretising the Poisson equation.
Let Ω be a non rectangular bounded domain. We embed Ω in a rectangular domain D with a uniform Cartesian grid so that grid points outside of the domain Ω are represented by Ω+. Now a Cartesian grid point x
0 is called regular if the
standard five point stencil centered at x0 is completely contained in the domain Ω.
At the regular grid point, the scheme (1.0.1) is complete in the sense that all the five solutions are part of the unknown [42, 43, 44, 65, 70]. Otherwise such a central grid pointx0 is called irregular and the scheme (1.0.1) centered atx0 is not complete since
some of the five solutions are contained in Ω+ which are therefore not known. Our
task is therefore to complete the scheme at irregular grid pointsx0.
We complete the scheme (1.0.1) at the irregular grid point x0 by the method of
continuation of solution using local approximations based on the so-called immersed interface idea. We proceed through the following steps:
• We orthogonally project an irregular grid point x0 onto the boundary Γ
to form a projected pointx∗ on Γ and then form a local coordinate system
(ξ, η) centered at x∗;
• Next, we represent the unknown solution by a second order Taylor’s ex-pansion in local coordinates about the projected pointx∗;
• We then use the following conditions to determine the Taylor’s expan-sion:
• Equate the solution described by the Taylor’s expansion at the irregular grid pointx0 tou0and treatu0 as known at this stage,
• Equate the equation at the projected point x∗,
• Use the boundary condition and its tangential derivatives atx∗,
Γ to the solution in terms of the Taylor’s expansion;
• We then use the completely determined Taylor’s polynomial to extend the solution beyond Γ locally and approximate the solutions at grid points in Ω+needed to complete the scheme (1.0.1) atx
0 in a second order manner.
In order to carry out the above steps, we use level set approach as detailed in Sections 2.1 and 2.2. Thus Γ is identified with the zero level set of the level set function.
As a consequence of our approach for completing the scheme (1.0.1) at all the irregular grid points for the domain, the resulting finite difference system matrix remains symmetric positive definite and maintains the sparsity of the standard second order scheme. That is, only the diagonal entries of the system matrix corresponding to the irregular grid pointsx0are modified according to the geometry of the boundary.
As well, the associated right hand side is modified based on geometry of boundary, boundary condition and forcing function atx∗.
Our second order method is very efficient and its implementation is easy to follow. Moreover, we have extended and carried it out for general boundary conditions and our treatment handles the general boundary conditions in two and three dimensions without much difficulties. In fact, the second order method has been extended and carried out for the Neumann and Robin boundary conditions and extensions to three dimensions have also been carried out as detailed in the next Chapter.
costs.
We now follow up with what needs to be done to develop the fourth order method. We use the following standard compact nine point finite difference scheme
1 h2{−
10 3 ui,j+
2
3(ui,j+1+ui,j−1+ui+1,j+ui−1,j) +1
6(ui+1,j+1+ui+1,j−1+ui−1,j+1+ui−1,j−1)} = 8
12fi,j+ 1
12{fi+1,j+fi−1,j+fi,j+1+fi,j−1} (1.0.2) to replace the standard five point stencil scheme (1.0.2). The outlined steps for the second order method are modified accordingly for the fourth order method. For example in step one, some grid points which are regular for the second order method are now irregular for the fourth order method as detailed in Section 3.3. Also in step two, we represent the unknown solution by a fourth order Taylor’s expansion about the projected points x∗ and the details are carried out in Section 3.2. Other
modifications include determining the projection pointsx∗ in a fourth order manner.
The numerical tests conducted show a much improvement in terms of the mag-nitude of errors over our second order method. The resulting system matrix is also symmetric positive definite maintaining the sparcity of the fourth order standard com-pact nine point finite difference scheme (1.0.2) used. We describe the model problem of interest in the next section.
1.1
The Model Problem
Let Ω be an open bounded subset of Rk with a smooth boundary ∂Ω for k= 2 or 3.
Taking the domain Ω to be non rectangular, consider the following boundary value problem of the elliptic type
(
−div(D· ∇u) +cu =f in Ω
negative. Dis in general thedif f usion tensor which is a positive definite symmetric matrix and f is the source of the diffusion withg as the boundary value.
1.1.1
Physical Background of Diffusion Processes
Diffusion of the mass of a substance may be described as a physical process that equilibrates concentration differences of the substance without creating or destroying mass. The physical observation of diffusion can easily be cast in a mathematical formulation.
According to the F ick0s law, the equilibration property of diffusion is expressed
by the flux law [11],
j =−D· ∇u ∀x∈S, (1.1.2)
where S is any surface in the domain Ω, j is the flux or flow of the mass of the substance across S and D is called diffusion tensor. That is, the concentration gra-dient ∇u causes the flux j which aims to compensate for this gradient. Diffusion is described as isotropic when the diffusion tensor D and the flux j are parallel in which caseDmay be replaced by a positive scaler valued diffusivityκ. In the general anisotropic case, j and ∇uare not parallel. The observation that mass is not created or destroyed is expressed by the continuity equation
divj = 0 ∀x∈Ω. (1.1.3)
Suppose that there is an external input f0 to the concentration of the diffusion and
a growth potential or growth rate of the mass of the substance c0 at x ∈ Ω. Then
(1.1.3) becomes
divj +c0u=f0 ∀x∈Ω. (1.1.4)
Plugging (1.1.2) into (1.1.4), we obtain the pointwise equation
−div(D· ∇u) +c0u=f0, x ∈Ω. (1.1.5)
before, the diffusion tensor is replaced by a positive constant κ(x) ≡ κ in Ω and (1.1.5) reduces to
−4u+cu=f, x ∈Ω, (1.1.6)
wheref =f0/κ, c=c0/κ and (1.1.6) is referred to as the steady state heat equation
[11, 56].
We consider the usual boundary conditions which arise from the model: (a) P rescribed distribution of temperature on the surf ace ∂Ω. This implies
u=g on ∂Ω = Γ, (1.1.7)
where g is a given function on ∂Ω. This kind of boundary condition is the dirichlet type;
(b)P rescribed f low of heat across the surf ace ∂Ω. Using Fick’s law, we have −κ∂u
∂ν =g on Γ. (1.1.8)
(c)Newton0s law of Radiation. If the heat flux across the surface is proportional
to the difference in temperature between the surface and the surrounding medium, then by the Fick’s law, we have
−κ∂u
∂ν =γ(u−u0) on Γ, (1.1.9)
where u0 is the temperature of the surrounding medium. This is the Robin(Mixed)
boundary condition.
Based on the boundary measurements obtained from any one of the boundary con-ditions (1.1.9) or (1.1.8) or (1.1.7) together with the pointwise heat equation (1.1.5), our model of study (1.1.1) is obtained.
The Porous Medium Equation
Consider an isentropic gas flow in a homogeneous porous medium. Then its normal-ized density distribution umay be described by a non steady state diffusion equation of the form (1.1.1) where D is given by [8]
D=mum−1, u≥0 and m= 1 +α≥2. (1.1.10) This equation arises in many applications including the theory of ionized gases where m > 1 and in plasma physics with m <1. Another application is groundwater flow which is the lateral flow of water in the deep aquifer zones where all interstices are filled with water under pressure greater than the atmospheric pressure. This body of water eventually discharges into streams, rivers, lakes and the oceans. The discharge distribution is controlled by the variations in flow resistances and consequently the biochemical processes constituting wetland ecology are distributed [31, 86].
For the groundwater flow in an unconfined two dimensional anisotropic aquifer, u represents the height of the phreatic surface above and the symmetric matrixD is defined to be the vertical average of the hydraulic conductivity matrixK(x, y, z), [13]
D(u) = 1 u
Z u
0
K(x, y, z)dz. (1.1.11)
Other examples of the form (1.1.1) include the density distribution of a biological population, the potential of fluid field flows, the equilibrium configuration of an elastic membrane, the potential of an electric field and other numerous examples in [11, 56, 95].
1.2.1
Time Dependent Diffusion Problems
We consider various kinds of linear diffusion problems cast in the form
∂u
∂t =div(κ(x)∇u) in Ω×(0, T),
u(0) =u0
Bu =v(t, x) ∀x∈∂Ω = Γ,
(1.2.1)
where the domain is irregular and B represents the boundary operator. Supposing for simplicity that κ(x) = κ is a constant, we extend our fourth order method to efficiently semi-discretize (1.2.1) in spatial domain incorporating the given boundary values. That is, we discretize (1.2.1) in space to obtain the following form of an ordinary differential equation
( Qdu
dt =Hu+Bv in Ω×(0, T),
u(0) =u0,
(1.2.2)
where Bv is seen as a source on the boundary Γ, H is the stiffness matrix and Q is the mass matrix.
Both of the resulting matrices H and Q from our fourth order method are sym-metric positive definite and sparse and so with a fourth order time discretization method, we solve (1.2.1) in a fourth order manner in space and time.
1.2.2
The Eigenvalue Problem
With Ω as an open bounded domain having a smooth boundary ∂Ω, consider the problem of determining the non-trivial solutions uk to the problem
−∆u =λu in Ω
Bu = 0 on ∂Ω, (1.2.3)
and a corresponding eigenvalue λk ∈ R. The boundary operator B is either of the
If (1.2.3) has nontrivial weak solutionsukcalled eigenfunctions, then the constants
λk which form an increasing sequence
0< λ1 ≤λ2 ≤λ3 ≤ · · · ≤λk· · · → ∞ as k→ ∞ (1.2.4)
are calledeigenvalues of the negative Laplace operatorwith Dirichlet boundary con-ditions or the Robin boundary concon-ditions respectively [35, 47, 79]. In other words, trivial solution exists for all values of the parameter λ in (1.2.3) but nontrivial solu-tions u exist only for eigenvalues λ.
Eigenfunctions and eigenvalues are very essential in mathematics and science in that eigenfunctions and eigenvalues often have physical interpretations in physical problems. For example,
• In vibration problems, the square of the eigenvalues are proportional to the natural frequencies of the system and the eigenfunctions are the shapes of the normal mode vibrations. The normal mode is a motion where all parts of a system vibrate with simple Harmonic motion with the same frequency ω but possibly different amplitudes, and all parts go through the equilibrium position together [87].
• In the case of buckling of a column, the eigenvalues determine the loads at which the column buckles.
• Also in quantum mechanics, the eigenvalues represent the only possible measurable values of the energy levels of the physical system.
1.2.3
Application to Boundary Control
Suppose in a thermal treatment of a body Ω within a chamber, we want to maintain the temperature within a part ¯Ω of the body at ¯u(t, x). Such temperature distribution within Ω is be governed by the diffusion equation (1.2.1). So for such a desired temperature distribution ¯u(t, x), we might try to choose a suitable boundary value v so that the solution u(t, x) of (1.2.1) is close to ¯u(t, x) in ¯Ω.
In such a problem, we refer to (1.2.1) as a control system for it represents a way of determining u by v under proper boundary conditions. Assume that we want to achieve the goal of making u as close as possible ¯u in an optimal way, then this objective is formulated as an optimal control problem subject to the control system (1.2.1). Thus a criterion for measuring the performance of the control system called a cost functional is specified as
J(v) = Z T
0
Z
¯ Ω
|u(t, x)−u(t, x)|¯ 2dxdt+
Z T
0
Z
∂Ω
|v(t, x)|2dsdt. (1.2.5)
The goal is to minimize the aboveJ(v) by choosingv properly. The first term inJ(v) is the cost of not attaining the equilibrium temperature distribution and the second term is the cost of the control effort.
Using our nine-point compact fourth order method discretization, we design an optimal state feedback stabilization [18, 29, 39, 81] of the Dirichlet boundary control problem for the heat equation in Chapter 5.
1.2.4
Extension to Elliptic Interface Problems
The interface problem is defined by the elliptic equation of the form −∇ ·(β(x)∇u) +c(x)u = f(x) x∈Ω
∂u
the diffusion coefficient β is piece-wise constant over Ω and Dirichlet boundary con-ditions are given on ∂Ω. Thus, we define the interface problem such that β is given by
β(x) =
β+, x∈Ω+
β−, x∈Ω−
, (1.2.7)
where β± are positive constants,
Ω− ={x∈Ω :φ(x)<0}, Ω+ ={x∈Ω :φ(x)>0}, Γ ={x∈Ω :φ(x) = 0} (1.2.8) with φ as a level set function and Ω = Ω+∪Ω−∪Γ.
Using our fourth order method we develop an M-matrix nine-point compact finite difference method to solve the interface problem over the whole domain across the material interface.
1.3
Other Related Work
The problems of our present interest in this work and related ones have extensively been investigated. We describe some of the relevant research.
at a distance √2h away. A linear combination of the two finite difference schemes produces the fourth order accurate method.
A second order accurate finite difference scheme using finite volume discretization was developed in [43] to solve the variable Poisson equation with Dirichlet boundary conditions in one and two dimensions. Their approach embeds the domain in a regular Cartesian grid and uses the flux at the cell boundaries. In two dimensions, the boundary is approximated by straight line segments. In order to evaluate the flux, the normal direction is determined at the center point of each boundary segment. Then quadratic extrapolation of the solution values at two intersection points of the normal with selected Cartesian grid lines and the boundary value on the normal is determined and the derivative at the cell boundary approximates the flux. A non symmetric linear system is therefore resulted.
In [65], the basic idea of the ghost fluid method was employed to develop a first-order accurate symmetric finite difference scheme based on the Cartesian grid to solve the variable coefficient Poisson equation in the presence of an irregular interface. Subsequently in [34], the approach in [65] was modified to obtain a second-order accurate symmetric finite difference scheme based on Cartesian grid to solve a variable coefficient Poisson equation with Dirichlet boundary conditions. The modification used the signed distance level set function to obtain a linear interpolation from the boundary value and the solution values in coordinate-wise directions which determines the ghost fluid values.
1.4
Outline of Thesis
In Chapter 2, we develop our second order method for the Poisson equation on non-rectangular two-dimensional domains with the Dirichlet boundary conditions. We then describe the extension to the general boundary conditions and then to non-rectangular three-dimensional domains. The implementation in two dimensions with the use of level set functions is then described. Numerical results for Dirichlet and Neumann boundary conditions are presented for two dimensions. The second order-ness of the proposed method is confirmed through numerical tests and least square estimates for the rate. The well-posedness is discussed and the convergence rate established using the discrete maximum principle.
In Chapter 3, we establish the compact nine-point stencil scheme on a square domain using hat functions as test functions through the weak formulation of the Poisson equation. We then extend our second order method to obtain the fourth order method for the Poisson equation on non-rectangular domains with general boundary conditions. The fourth orderness of the method is confirmed through numerical tests and least square estimates for the rate.
In Chapter 4, we extend the 9-point compact fourth order method to discretize the associated parabolic PDE in space resulting in an ODE system which presumably con-tains the dominant characteristics of the PDE . The validity of this semi-discretization is established through the study of the associated eigenvalue problem. As such we present the estimates of the eigenvalues of the circular domain as against the exact eigenvalues. Consequently, we present the eigenvalues and eigenfunctions of other interesting domains.
loop systems are compared.
Finally we develop the nine-point compact fourth order method to solve the vari-able coefficient elliptic equation through least square optimization technique in Chap-ter 6. That is, we develop an M-matrix 9-point compact method for the general elliptic equation. We also solve the interface problem where there is a material property dis-continuity in the domain. Specifically, consider the case where the diffusion coefficient is piece-wise constant. Numerical results are presented.
The Second Order Method
In this chapter we develop our second order finite difference method based on uniform Cartesian grids for solving elliptic equations of the form
−div(κ∇u) +cu =f in Ω
Bu =g on ∂Ω = Γ, (2.0.1)
on irregular boundary domains Ω. The method captures general boundary conditions for one, two and three dimensional problems but specifically we develop the method for two and three dimensions. We assume that Ω is open, bounded and has a general shape and also that the boundary Γ, the boundary value g and the source function f are smooth enough. The boundary operator B refers to any of the three different kinds of boundary conditions being the Dirichlet, the Newmann or the Robin(mixed) type.
For the purpose of developing the method, we take the diffusivity of the substance κ(x), to be normalized to unity and the growth potential of the substance c(x), to be zero. In the general case of using uniform Cartesian grid finite difference method to solve (2.0.1), the boundary Γ does not align with the grid lines but rather cuts between grid points. Thus, for a grid point x0 near Γ, a stencil of the standard
central difference scheme centered at x0 involves grid points with unknown solutions
and grid points with solutions outside of Ω. The equation (2.0.1) is not valid outside of Ω and so the grid points outside of Ω have solutions which are not known. Such a
standard central difference scheme cannot be applied directly on the stencil centered at x0 and so the finite difference scheme is said to be incomplete for that stencil. A
careful treatment of the solution in the neighborhood of the boundary is therefore necessary in order to complete the scheme at such points x0 to achieve the desired
accuracy. We accomplish such a task by the method of continuation of solution from the inside of the domain to the outside based on the boundary value, the geometrical properties of the boundary Γ and the equation. The essential part of the method is to capture the boundary geometry and hence capture the boundary values in the solution continuously.
In developing the method, we embed the domain Ω in a rectangular domain, say D= [a, b]×[c, d] with a regular uniform Cartesian grid. Then we represent the domain Ω by a level set function φ such that the zero level set of φ defines the boundary Γ as below:
Γ := {(x, y) :φ(x, y) = 0}. (2.0.2) Consequently, we use the level set function representation to make the following definitions:
Ω = {(x, y) :φ(x, y)<0} Ω+ = {(x, y) :φ(x, y)>0}
D = Ω∪Ω+∪Γ
(2.0.3)
such that the solutions at grid points contained in Ω+ are not known while the
solutions at grid points in Ω are the unknowns.
2.1
Second Order method in Two Dimensions
We begin by considering the Poisson equation −∆u =f in Ω
Bu =g on Γ, (2.1.1)
on an irregular domain Ω which is two dimensional in space. This is obtained from (2.0.1) by assuming a zero growth potential and a constant coefficient of diffusivity.
As explained above, we use the standard central difference scheme 4u0−u−1−u−2−u1−u2
h2 =f0 (2.1.2)
which is second order accurate on a standard five point stencil to develop our second order method for solving (3.2.1) where f0 is the evaluation of f at the central grid
point.
In order to explain the method in two dimensions, we illustrate a portion of the computational domain intersecting with the boundary Γ and part of the domain Ω in Figure 2.1. Figure 2.1 shows a typical situation where the boundary cuts through the grid lines in a standard five-point stencil and as noted in Section 2.1, we develop our methods to complete the scheme (2.1.2) for such a stencil.
For the five-point stencil in Figure 2.1 centered at x0, the solutions u0, u−1 and
u−2 are part of the unknowns in Ω while the solutions u1 and u2 are outside of Ω and
so are not known. Therefore the scheme (2.1.2) cannot be used directly to discretize the Poisson equation at x0 since the stencil involves solutions outside of the domain
Ω. As a result, the scheme (2.1.2) is classified as incomplete for the standard stencil centered atx0 and the grid pointx0 is called irregular. Otherwise, if all the solutions
at the grid points of the 5−point standard stencil belong to Ω, the scheme (2.1.2) is complete for the stencil and the central grid point is called regular.
Now we describe how to complete the scheme at the irregular grid points x0. We
first project each irregular grid point x0 onto the boundary orthogonally to obtain
Figure 2.1: A portion of the boundary intersecting 5−point standard stencil.
(x
1,u1)
(x
2,u2)
(x*,u*)
(x*
1,u
*
1)
(x
−1, u−1)
(x
−2,u−2)
Boundary Γ (x
0,u0)
(x*
2,u
*
2)
(x
5,u5)
(x
6,u6)
(x
3,u3)
(x
4,u4)
manner. We calculate these projected points on Γ by using the grid point values of the level set function. Thus a set of pairs of irregular grid points and their projected points are obtained. These projected points are multi-directionally calculated from the irregular grid points and are not necessarily on grid lines.
As noted in Section 2.1, by the method of continuation of the solution, we extend the unknown solution u beyond the boundary locally so that the scheme (2.1.2) can be supported in a second order manner at all the irregular grid points. Thus we obtain second order approximation of the solutions at the grid points in Ω+ needed
to complete the scheme at the irregular grid points. In this way the scheme (2.1.2) is complete for the whole domain.
In order to compute the continuation extension, we introduce a local coordinate transformation [26] given by
ξ= (x−x∗)α
xξ+ (y−y∗)αyξ
η= (x−x∗)α
xη+ (y−y∗)αyη
, (2.1.3)
where (x∗, y∗) are thex−y coordinates of the projected point x∗ and α
the direction cosine between the x and local η coordinate directions.
Figure 2.2: An irregular grid point with its projected point and the local coordinate transformation.
x
0
x* in xy−coordinates
(0,0) in ηξ local coordinates
y−direction
x−direction
η−direction as
tangential direction
ξ direction as
normal direction
Γ
ξ
η
Figure 2.2shows the projection of an irregular grid pointx0onto Γ at the projected
point x∗ and the local coordinate system where the tangential direction to Γ at x∗
is depicted by η and the normal by ξ. We then illustrate the new local coordinate system formed at x∗ in Figure 2.3.
We next use Taylor’s 2ndorder expansion to represent the unknown solution uin the local neighborhood of the projected point x∗ in local coordinates by
u(ξ, η) = u∗+u
ξξ+uηη+
1 2uξξξ
2+u
ξηξη+
1 2uηηη
2, (2.1.4)
where u∗ is the boundary value of the unknown solution at the projected point x∗.
We intend to use (2.1.4) to calculate the approximation to the solutions at the grid points in Ω+ needed to complete the scheme at the irregular grid points in a
second order manner. In order to use (2.1.4), we need to determine the six unknown coefficients
u∗, u
Figure 2.3: Local boundary representation of Γ at the projected point.
ξ−axis
(0,0)
η−axis
boundary Γ
χ(0)=0, χ’(0)=0 & κ=χ"(0)
ξ=χ(η)
at the projected points and therefore six conditions are needed.
In the first place, since the local coordinate transformation is invariant of the equation, the Poisson equation is written in local coordinates as
−(uξξ+uηη) = f∗ atx∗. (2.1.5)
which is considered as one condition. Thus we use (2.1.5) to eliminate uξξ in (2.1.4)
to simplify the Taylors expansion to
u(ξ, η) = u∗+uξξ+uηη+uξηξη+ 1
2uηη(η
2−ξ2)− 1
2f
∗ξ2 (2.1.6)
with five unknown coefficients.
The local boundary representation of Γ at the projected points is illustrated by Figure 2.1 and we represent Γ in the neighborhood of the projected point x∗ by
where χ(η) is be continuously differentiable with κ as its curvature at x∗. Thus the
local regularity ofχ(η) atx∗ is assumed for the regularity of Γ atx∗ as in Figure 2.1.
We now appeal to the boundary data to determine the remaining five unknown coefficients in (2.1.7). We continue with the description by considering the Dirichlet boundary conditions.
2.1.1
Dirichlet Boundary Conditions
Suppose that the Dirichlet boundary conditions are given on Γ. Then we have another condition needed for the second order Taylor’s expansion as
u∗(0,0) = g(x∗(s)), (2.1.8) where (η, ξ) are the local coordinates centered atx∗ on Γ andsis a boundary
param-eter.
Four more conditions are needed in order to determine the Taylors expansion and then utilize it for approximating the solution at grid points in Ω+needed to complete
the scheme (2.1.2) at irregular grid points. We describe two ways for achieving that: • The first approach is based on the fact that the tangential derivative
uη(0,0) at x∗ may be calculated from (2.1.8).
That is, by definition, we have uη(0,0) :=
∂g ∂η(x
∗(s)) =g
η(x∗(s)) (2.1.9)
which constitutes an additional condition.
Differentiating (2.1.9) with respect to η and then evaluating at x∗, we
obtain
uηη+κuξ =gηη(x∗(s)) (2.1.10)
Therefore, substituting (2.1.8) and (2.1.9) into (2.1.6), the unknown coef-ficients in the Taylor’s expansion reduce to three in
u(ξ, η) =g+uξξ+gηη+uξηξη+
1 2uηη(η
2 −ξ2)− 1
2f
∗ξ2, (2.1.11)
where the remaining unknown coefficients are uξ, uξη, uηη.
At this point we have used the equation and the boundary measurement atx∗ in the Taylor’s local expansion of the unknown in the neighborhood
of x∗.
The following steps show the uniqueness of our method where we make use of the differential properties of Γ and the Dirichlet boundary values at the nearest projected points on Γ. Three more conditions are needed in order to completely describe (2.1.11). In the first place, we equate the Taylor’s expansion at the irregular grid point as in
u0 =g+uξξo+gηηo+uξηξoηo+
1 2uηη(η
2
o −ξo2)−
1 2f
∗ξ2
o. (2.1.12)
We then evaluate the expansion again at two nearest projected points to the main projected pointx∗ on Γ as shown below:
gi∗ =g+uξξi+gηηi+uξηξiηi+
1 2uηη(η
2
i −ξi2)−
1 2f
∗ξ2
i, (2.1.13)
where (ξi, ηi) with i = 1,2 are the local coordinates of the two nearest
projected points to x∗ on Γ. Also g∗
i are the Dirichlet boundary values at
the two nearest projected points on Γ. We now have the conditions to be able to determine all the remaining unknown coefficients
uξ, uξη, uηη
We put the system of (2.1.12) and (2.1.13) in a matrix vector equation form as; A uξ uξη uηη =
u0−g∗
g∗
1 −g∗
g∗
2 −g∗
+
−gηηo+12f∗ξo2
−gηη1+12f∗ξ12
−gηη2+12f∗ξ22
, (2.1.14)
where u0 is the unknown at the irregular grid pointx0 and
A=
ξ0 ξ0η0 12(η20−ξ02)
ξ1 ξ1η1 12(η21−ξ12)
ξ2 ξ2η2 12(η22−ξ22)
. (2.1.15)
We obtain the unknown coefficients in the form uξ uξη uηη
=x1u0−x2, (2.1.16)
where x1 ∈R3×1, x2 ∈R3×1 are given by
x1 =A−1
1 0 0
and x2 =A−1
−g∗
ηηo+ 12f∗ξo2−g∗
−g∗
ηη1+ 12f∗ξ12+g1∗−g∗
−g∗
ηη2+ 12f∗ξ22+g2∗−g∗
. (2.1.17)
Once the unknown coefficients in the Taylor’s expansion are determined, the solutions at grid points outside of Ω which are needed to complete the scheme (2.1.2) at the irregular grid points are evaluated using the resulting quadratic polynomial extension of u on the stencil.
Referring to Figure 2.1, the standard five point stencil centered at x0
has u1 and u2 outside of Ω. Using their coordinates based on the local
coordinate system at x∗, we evaluateu
1 and u2 by
uj =g+gηηj −
1 2f
∗ξ2
j +
³
ξj ξjηj 12(ηj2−ξj2)
´
as a function of the unknown u0. We put (2.1.18) in the form
uj =Aju0+Fj, j = 1,2 (2.1.19)
where
Aj =
³
ξj ξjηj 12(η2j −ξj2)
´ ·x1,
Fj =g +gηηj−
1 2f
∗ξ2
j +
³
ξj ξjηj 12(ηj2−ξj2)
´ ·x2.
In completing central difference scheme at the irregular grid point, (2.1.19) is substituted into (2.1.2) and as a consequence
X
j=1
³
ξj ξjηj 12(ηj2−ξj2)
´
·x1 (2.1.20)
modifies the diagonal entry of the system matrix corresponding to the unknown solutionu0 at the irregular grid point and
X
j=1
g+gηηj−
1 2f
∗ξ2
j +
³
ξj ξjηj 12(ηj2−ξ2j)
´
·x2 (2.1.21)
modifies also the associated right hand sidef0 of the system.
• The second approach is to avoid the calculation of the tangential deriva-tives gη in (2.1.9) at x∗. The overall accuracy of the method depends
on the accurate determination of the tangential derivatives. This is very technical and so instead, we treat gη and gηη as unknowns. Thus, we
avoid (2.1.9) in the first approach and then select one more closest pro-jected point on the boundary to accomplish the task of determining the unknown coefficients in the Taylors expansion. As a result, the Taylors expansion is written as
u=g+uξξ+uηη+uξηξη+
1 2uηη(η
2−ξ2)− 1
2f
∗ξ2 (2.1.22)
with the following four unknowns
As in the first approach, we solve for the unknown coefficients as a function of the unknown solution u0 in the form
uη uξ uξη uηη
=x1u0−x2, (2.1.23)
where x1 ∈R4×1, x2 ∈R4×1 are given by
x1 =A−1
1 0 0 0
, x2 =A−1
−g∗+ 1 2f∗ξo2
g∗
1−g∗+ 12f∗ξ12
g∗
2−g∗+ 12f∗ξ22
g∗
3−g∗+ 12f∗ξ32
(2.1.24) and A=
η0 ξ0 ξ0η0 21(η02−ξ02)
η1 ξ1 ξ1η1 21(η12−ξ12)
η2 ξ2 ξ2η2 21(η22−ξ22)
η3 ξ3 ξ3η3 21(η32−ξ32)
. (2.1.25)
Consequently, we evaluate u1 and u2 as in Figure 2.1 by
uj =Aju0+Fj, j = 1,2 (2.1.26)
where
Aj =
³
ηj ξj ξjηj 21(η2j −ξj2)
´
·x1, (2.1.27)
Fj =g−
1 2f
∗ξ2
j +
³
ηj ξj ξjηj 21(η2j −ξj2)
´
·x2. (2.1.28)
By substituting (2.1.26) into the finite difference scheme (2.1.2) to com-plete it at the irregular grid points,
X
j=1
³
ηj ξj ξjηj 21(ηj2−ξj2)
´
modifies the diagonal entry of the system matrix corresponding to the unknown u0 at the irregular grid point and
X
j=1
g−1 2f
∗ξ2
j +
³
ηj ξj ξjηj 21(η2j −ξj2)
´
·x2 (2.1.30)
also modifies the associated right hand side of the system.
Based on the curvature of Γ at the projected point, the local system matrices in (2.1.16) and (2.1.23) may be singular. The singularity which will be considered later occurs when the curvature of part of Γ containing x∗ and a closest projected point is
zero.
In either case, the scheme (2.1.2) is complete for the whole domain and that: • The Resulting system matrix A remains symmetric positive definite and
maintains the sparsity of the second order finite difference scheme (2.1.2). • Only the diagonal entries of the system matrix A are modified based on
the geometry of boundary.
• As well the right hand side is modified based on geometry of boundary, boundary condition and forcing function atx∗.
In the next Section, we consider the situation with the Neumann boundary con-dition.
2.1.2
Neumann Boundary Conditions
Given the Neumann boundary conditions, then
un =g(x∗(s)), at x∗ (2.1.31)
where n is the unit outward normal to Γ at x∗.
By definition, the Neuman boundary condition in local coordinates is given as ∂u
∂n =
(−χ0,1)
p
1 + (χ0(η))2 ·(uη, uξ) = g(η)
which is rewritten as
uξ−χ0(η)uη =
p
Evaluating (2.1.32) at the projected point x∗ where η= 0 andξ = 0 we have,
uξ(0,0)−χ0(0)uη(0,0) =
p
1 +χ0(0)g(0),
and therefore
uξ =g. (2.1.33)
We differentiate (2.1.32) with respect to η to obtain uξη +uξξχ0−χ00uη −χ0(uηη+χ0uξη)) =g0(η)
p
1 +χ02+g(η)pχ0χ00
1 +χ02 (2.1.34)
and evaluate at (0,0) in local coordinates to the condition
uξη −κuη =gη. (2.1.35)
We have the two treatment approaches again and we continue the description with the first where the tangential derivatives at x∗ may be calculated numerically. By
substituting (2.1.33) and (2.1.35) into (2.1.6), the Taylor’s expansion reduces to u=u∗+gξ+g0ξη+u
η(η+κηξ) +
1 2uηη(η
2−ξ2)−1
2f
∗ξ2. (2.1.36)
with the following three unknowns
u∗, u
η, uηη.
In order to determine these three unknown coefficients, we need three conditions and the first of them as before is to equate (2.1.36) to u0 at the irregular grid point x0
which is given as
u0 =u∗ +gξ0+g0ξ0η0+uη(η0+κη0ξ0) +
1 2uηη(η
2
0 −ξ02)−
1 2f
∗ξ2
0. (2.1.37)
Now, (2.1.36) is an expression of the unknownuin the neighborhood of the projected pointx∗and so we compute the Neumann boundary condition at the nearest projected
points x∗
i by
∂u ∂ni
(x∗i) = ∂u ∂ξ(x
∗
i)
∂ξ ∂ni
+∂u ∂η(x
∗
i)
∂η ∂ni
where ni is the unit outward normal to Γ at an arbitrary point x∗i on Γ.
To accomplish the task of determining the unknowns coefficients, we employ (2.1.38) at two selected closest projected points to x∗ on Γ. We thus obtain
∂u ∂ni
(x∗
i) = αniξ(g+gηηi+uηκηi−uηηξi−f
∗ξ
i)+αniη(gηξi+uη+uηκξi+uηηηi), (2.1.39)
where αniξ is the cosine of the angle between the local unit outward normals at x
∗
i
and x∗ and α
niη is the cosine of the angle between the local unit tangent vector at x
∗
i
and the local normal at x∗.
The quantities αniξ and αniη reveal curvature properties of Γ between two
pro-jected points such that if there is a zero curvature, then αniη is zero and αniξ is
approximately one in which case a form of treatment to be described later has to be performed.
We now put the linear system comprising of (2.1.37) and (2.1.39) into a matrix vector equation as :
A u∗ uη uηη = u0 0 0 +
gξ0+g0ξ0η0−12f∗ξ02
αn1ξ(g+g
0η
1−f∗ξ1) +αn1ηg
0ξ
1− ∂u∂n(η∗1, ξ1∗)
αn2ξ(g+g0η2−f∗ξ2) +αn2ηg0ξ2− ∂u
∂n(η∗2, ξ2∗)
, (2.1.40)
where A=
1 η0+κη0ξ0 21(η20−ξ02)
0 αn1η(1 +κξ1) +αn1ξκη1 (αn1ηη1−αn1ξξ1)
0 αn2η(1 +κξ2) +αn2ξκη2 (αn2ηη2−αn2ξξ2)
. (2.1.41)
We then write the solution as u∗ uη uηη
where x1 =A−1 1 0 0 and
x2 =A−1
gξ0+g0ξ0η0−12f∗ξ02
αn1ξ(g+g
0η
1−f∗ξ1) +αn1ηg
0ξ
1−∂u∂n(η∗1, ξ1∗)
αn2ξ(g+g0η2−f∗ξ2) +αn2ηg0ξ2− ∂u
∂n(η∗2, ξ2∗)
. (2.1.43)
We now calculate the solution at the grid points in Ω+ using (2.1.42) by
uj =gξj+gηξjηj−
1 2f
∗ξ2
j +
³
1 ηj+κξjηj 21(ηj2−ξj2)
´
(x1u0−x2) (2.1.44)
as a function of the unknown u0. We put (2.1.44) in the form
uj =Aju0+Fj, j = 1,2 (2.1.45)
where
Aj =
³
1 ηj+κξjηj 21(ηj2−ξj2)
´
·x1 (2.1.46)
and
Fj =gξj +gηξjηj −1
2f
∗ξ2
j +
³
1 ηj+κξjηj 21(ηj2−ξj2)
´
·x2. (2.1.47)
In completing the central difference scheme at the irregular grid point, (2.1.45) is substituted into (2.1.2) and as a consequence
X
j=1
³
1 ηj +κξjηj 21(η2j −ξj2)
´
·x1 (2.1.48)
modifies the diagonal entry of the system matrix A corresponding to unknown u0 at
the irregular grid pointx0 and
X
j=1
gξj +gηξjηj −
1 2f
∗ξ2
j +
³
ξj ξjηj 12(η2j −ξj2)
´ ·x2
The second approach is to treat gη as an unknown and use the next closest
pro-jected point on Γ as a condition to replace the numerical calculation of gη.
Hence the scheme (2.1.2) is complete for the whole domain. Since the Neumann boundary condition treatment is a variant of the Dirichlet boundary condition treat-ment, the results in the previous section 2.1 applies for this Section in principle.
We continue with the second approach for treating the Robin boundary condition in the next Section.
2.1.3
Robin Boundary Conditions
When the given boundary condition is of the Robin type then
un+au=g(x∗(s)), on Γ (2.1.49)
where n is the unit outward normal to Γ and a >0. This implies that (−χη,1)
p 1 +χ2
η
·(uη, uξ) +au=g(η) (2.1.50)
and at the projected point x∗, we have
uξ+au=g. (2.1.51)
substituting (2.1.51) into (2.1.6), we have u(ξ, η) =u∗(1−a∗ξ) +gξ+u
ηη+uξηξη+
1 2uηη(η
2−ξ2)− 1
2f
∗ξ2. (2.1.52)
with four unknowns
u∗, uη, uξη, uηη.
As a complimentary to the first approach of the previous Section, we consider the second approach. We equate (2.1.52) at the irregular grid point to obtain
u0 =u∗(1−a∗ξ0) +gξ0 +uηη0+uξηξ0η0+
1 2uηη(η
2
0 −ξ02)−
1 2f
∗ξ2
Then we select three closest projected points on Γ and substitute (2.1.52) into (2.1.49) to obtain
g∗
i = αξ ~~ni(g−a∗u∗+uηξηi−uηηξi−f∗ξi) +α~η ~ni(uη+uηξξi+uηηηi)
+a∗
i(u∗(1−a∗ξi) +gξi+uηηi+uξηξiηi+ 12uηη(ηi2−ξi2)− 12f∗ξi2)
= u∗(a∗
i(1−a∗ξi)−a∗αξ ~~ni) +un(α~η ~ni+a
∗
iηi)
+uηξ(α~η ~niξi+α~ξ~nηi+a∗iξiηi) +uηη(α~η~nηi
−αξ~n~ ξi +a
∗
i
2 (ηi2−ξi2)) +αξ~n~ (g−f∗ξi) +a∗i(gξi− 12f∗ξ2i),
(2.1.54)
where (ξi, ηi) for i = 1,2,3 are the coordinates of the closest projected points to x∗
on Γ based on the coordinate system atx∗ and α
~
ξ ~ni is the cosine of the angle between
the normal directions at x∗ and x∗
i. Also α~η ~ni is the cosine of the angle between the
tangent direction at x∗ and the normal at x∗
i. The four unknowns
u∗, u
η, uξη, uηη
are solved by
A u∗ uη uξη uηη = 1 0 0 0
u0+
1
2f∗ξ02−gξ
g∗
1 −α~ξ ~n1(g−f∗ξ1)−a∗1(gξ1− 12f∗ξ21)
g∗
2 −α~ξ ~n2(g−f∗ξ2)−a∗2(gξ2− 12f∗ξ22)
g∗
3 −α~ξ ~n3(g−f∗ξ3)−a∗3(gξ3− 12f∗ξ32)
, (2.1.55)
where A is given by
A =
(1−aξ0) η0 η0ξ0 12(η02−ξ02)
a∗
1(1−a∗ξ1) a1∗η1 a∗1ξ1η1 a
∗
1
2 (η21−ξ12)
a∗
2(1−a∗ξ2) a2∗η2 a∗2ξ2η2 a
∗
2
2 (η22−ξ22)
a∗
3(1−a∗ξ3) a3∗η3 a∗3ξ3η3 a
∗
3
2 (η23−ξ32)
+
0 0 0 0
−a∗α
~
ξ ~n1 α~η ~n1 α~η ~n1ξ1+α~ξ ~n1η1 α~η ~n1η1−α~ξ ~n1ξ1
−a∗α
~
ξ ~n2 α~η ~n2 α~η ~n2ξ2+α~ξ ~n2η2 α~η ~n2η2−α~ξ ~n2ξ2
−a∗α
~
ξ ~n3 α~η ~n3 α~η ~n3ξ3+α~ξ ~n3η3 α~η ~n3η3−α~ξ ~n3ξ3
In line with the first approach, we differentiate (2.1.50) and evaluate at x∗ to
obtain
uξη =gη−(κ+a)uη. (2.1.56)
and then use (2.1.56) to reduce the system (2.1.55) by one dimension. We proceed as before to obtain the solutions at grid points in Ω+ to complete the scheme for the
whole domain.
2.1.4
Addition of a Potential term
We consider the case of a non zero growth potential c >0 where the elliptic equation (2.0.1) is then modified to
−∆u+cu =f in Ω
Bu =g on Γ . (2.1.57)
By incorporating the potential term, the standard second order five-point stencil finite difference scheme (2.1.2) is modified to
4u0−u−1−u−2−u1−u2
h2 +c0u0 =f0. (2.1.58)
The various treatments of the boundary conditions remain unchanged and the effect on the overall system is that the Identity matrix is added to the system matrix.
2.2
Second Order Method in Three dimensions
In three dimensions, we use the following 7−point standard central difference scheme 6u0−u−1−u−2−u−3−u1−u2−u3
h2 =f0 (2.2.1)
which is second order accurate to develop the method.
