3.2 Error estimation in the finite element method
3.2.2 The Dual Weighted Residual method
For the numerical experiments undertaken in this chapter the Dual Weighted Residual (DWR) approach has been selected, because it allows the estimation of the error in the approximation of a functional depending on the solution of the PDE. Such quantities, e.g. the lift or drag in fluid dynamics applications, and the error in their approximation, are of direct concern to engineers using the finite element method, while the energy norm of the error is more of a theoretical tool. Further, the use of this error estimation technique allows construction of meshes adapted to approximate the quantity of interest in an even more efficient way than on meshes adapted to produce minimal error in the energy norm. The DWR method is introduced in this section for the case of a linear functional and a linear PDE. Further details may be found in [11] and the references therein. Consider the derived quantity
J(u) := Z
Ω
gudΩ, (3.2.2)
where g is a kernel function and u ∈ H1gD,ΓDis the unknown solution to a PDE whose weak
form is
a(u, v) = b(v) ∀v ∈ H10,ΓD. (3.2.3)
Here H1f,ΓD denotesu ∈ H
1(Ω) : u= f on Γ
D , where ΓDis the (non-empty) Dirich-
let part of the boundary of the domain Ω. Let uh be defined as the solution of the FE
discretisation of the weak form (3.2.3), hence uh∈ VgD,h such that
a(uh, vh) = b(vh) ∀vh∈ V0,h, (3.2.4)
where Vf,h denotes the FE function space Vf,h⊂ H1f,ΓD, and let the discretisation error e
be defined as in (3.2.1). The dual in the Dual Weighted Residual method comes from its use of the solution z of the dual problem, find z ∈ H10,ΓDfor which
a(ϕ, z) = J(ϕ) ∀ϕ ∈ H10,ΓD, (3.2.5)
which is well defined since J is a bounded linear functional and we assume the forward problem (3.2.3) to be well defined. Furthermore the linearity of J implies that the error in the quantity of interest, J(u) − J(uh) = J(e), and from (3.2.5)
J(e) = a(e, z)
= a(u, z) − a(uh, z)
Note that the use of any approximation zh∈ V0,h for z in (3.2.6) does not provide any
information on J(e) since b(zh) − a(uh, zh) = 0 ∀zh∈ V0,h by Galerkin orthogonality. This implies that the FE approximation to the dual solution is not sufficient to provide a successful error estimator, at least so long as it is computed from the same space as uh
is. Of course computing an exact solution to the dual problem (3.2.5) is just as difficult as computing an exact solution to the original problem (3.2.3) and therefore it is neces- sary to consider an approximate solution for z, which implies that instead of the true error J(e) it is only possible to obtain an estimate for it. To emphasise that this approxima- tion is different from zhit will, from here on, be denoted by zapp, and the corresponding
approximation shall be denoted by
Jest:= b(zapp) − a(uh, zapp). (3.2.7)
Several ways of defining zapp in order to obtain such an estimate Jesthave been proposed,
e.g.
1. solve the dual problem with a higher order method on the same mesh, 2. solve the dual problem on a uniformly refined mesh, or
3. use the same trial space to solve the dual problem, but use a higher order interpolant of zhas zapp.
Solving the dual problem with a higher order method or on a uniformly refined mesh means that the error estimation will be more expensive to obtain than solving the original discretised equations. In the former case the overhead results from more degrees of free- dom and larger matrix stencils, as well as the need for more accurate integration formulae. In the latter case the number of unknowns in the discrete problem will be increased by a factor of four or eight (in 2d respectively 3d). However, it does have the advantage of us- ing the same order method, hence significant parts of the original code should be directly reusable. Furthermore, the estimated error Jest can be used to get an improved approxi-
mation to J, thus effectively delivering the accuracy that one would get if one solved the problem on the finer mesh.
The third alternative of using a higher order interpolant of zh as zapp is described as
a strong competitor in [11]. However, the quality of this approximation Jest relies on a
super-convergence property which in turn relies on uniform meshes and smoothness of the dual solution z. As this work concentrates on anisotropic meshes this approach is less suitable, as demonstrated by the results in Section 3.4.3of this thesis, where the second and third methods are compared for an anisotropic problem for which the meshes are
stretched accordingly. It is found that the robustness of the second method justifies its additional expense.
Error estimates such as the DWR estimate are commonly used to guide local mesh refinement in regions which contribute most to the estimated error. This may be achieved, for example, by evaluating the right-hand side of (3.2.7) separately on each element and then refining those elements with the largest contributions. This refinement is normally done by subdividing elements into smaller ones of the same or similar shape. In two di- mensions this typically involves division in two [14] or four [70] sub-triangles and for tetrahedra in three dimensions either two or eight sub-elements (as in [74] or [83] respec- tively).
For the remainder of this chapter attention is restricted to problems in two dimensions and h-refinement based upon sub-division of triangles into four child elements. This method of local refinement is referred to as isotropic local mesh refinement in order to distinguish it from other approaches where an element may be refined differently in dif- ferent spatial directions. Treatment of hanging nodes, which come about when an element is refined but a neighbouring element is not, is briefly discussed in Subsection3.4.2.1.