In this thesis, we have proposed **finite** **element** **methods** for solving elliptic and elasticity problems with interfaces using locally modified meshes. The locally modified mesh is easy to generate from a Cartesian mesh in which some nodal points near the interface are moved on the interface. For elliptic problems with interfaces, the **finite** **element** method using the modified mesh is compared with another modified mesh, locally enriched triangulations also based on a Cartesian mesh, in which new nodal points are added on the intersection between grid lines and the interface. The locally enriched triangulation is also easy to generate. Based on our numerical experiments both meshes lead to second order accurate solution for elliptic problems with interfaces. With the adaptation, the triangles are quasi-uniform and the mesh maintains the Cartesian structure. In particular, the Cartesian structure is useful when storing matrices and developing iterative solvers. Neither of these properties hold for the locally enriched meshes.

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points where we look for unknown parameters of solutions – such as feeds from which pressures and tensions are calculated. Therefore, by a solu- tion of the problem by the **finite** **element** method, it is needed to divide the continuum or the entity into the **finite** number of sub-areas – elements. [12] The best understanding of the **finite** **element** **methods** is by its practical application, which is known as **finite** **element** analysis. **Finite** **element** analysis is applied in mechanical engineering as a calculation tool for the realization of technical analysis [13]. It includes the use of technologies that divide the continuous problem into small, in- dividual parts [14]. **Finite** elements analysis is a good choice for analysis of problems in complex areas (such as automobiles or pipelines), when the area changes (by reaction of the parts in solid state with movable parts), when the required accuracy varies across the area, or the solution lacks fluen- cy. This method can be also used for modeling of conveyor belts and its properties and it helps also for the economic level of their operation, above all attention to causes which cause degradation and damage of the belt. For example, Honus et al. [5] presented an interesting application of the **finite** **element** modeling of the stress-strain states in conveyor belts. Lodewijks [15] [16] [17] dealt with possible ways of the **finite** **element** method applied to the detailed research of selected prob- lems of conveyor belts. Mazurkiewicz [18] pre- sented the use of **finite** **element** **methods** by the right material properties determination.

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Now, stabilized **finite** **element** **methods** for the Stokes problem have been criticized because of their need to set a constant, called the stabilization parameter. Some alternatives have been proposed to set up this constant among which is the enriching space approach. The idea is to enhance a pair of non stable **finite** **element** spaces and look for a discrete solution in the underlying augmented stable spaces through the standard Galerkin method. What drives the choice of additional basis functions is to fulfill the inf-sup condition without increasing the size of the corresponding linear system, i.e, without incorporating extra degrees of free- dom. A sufficient condition to respect the latter constraint is to perform static condensation

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Rarani et al. [9] used analytical and **finite** **element** **methods** for prediction of buckling behavior, including critical buckling load and modes of failure of thin laminated composites with different stacking sequences. A semi- analytical Rayleigh–Ritz approach is first developed to calculate the critical buckling loads of square composite laminates with SFSF (S: simply-support, F: free) boundary conditions. Then, these laminates are simulated under axially compression loading using the commercial **finite** **element** software, ABAQUS. Critical buckling loads and failure modes are predicted by both eigenvalue linear and nonlinear analysis.

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Abstract. The implementation of discontinuous Galerkin **finite** **element** **methods** (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist of power series or functionals of the solution variables. Thereby, the exploitation of symbolic algebra to express a given DGFEM ap- proximation of a PDE problem within a high level language, whose syntax closely resembles the mathematical definition, is an invaluable tool. Indeed, this then facilitates the automatic assembly of the resulting system of (nonlinear) equations, as well as the computation of Fréchet derivative(s) of the DGFEM scheme, needed, for example, within a Newton-type solver. However, even exploiting symbolic algebra, the discretisation of coupled systems of PDEs can still be extremely verbose and hard to debug. Thereby, in this article we develop a further layer of abstraction by designing a class structure for the automatic computation of DGFEM formulations. This work has been implemented within the FEniCS package, based on exploiting the Unified Form Language. Numerical exam- ples are presented which highlight the simplicity of implementation of DGFEMs for the numerical approximation of a range of PDE problems.

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Meshing of complex geometries remains a challenging and time consuming task in engi- neering applications of the **finite** **element** method. There is therefore a demand for **finite** **element** **methods** based on more flexible mesh constructions. One such flexible mesh paradigm is the formulation of **finite** **element** **methods** on composite meshes created by letting several meshes overlap each other. This approach enables using combinations of meshes for certain parts of a domain and reuse of meshes for complicated parts that may have been difficult and time consuming to construct.

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An alternative is to assume that the solution is harmonic in time. This removes the time derivative from the equation, replacing it with the frequency of the solution instead. The time- harmonic Maxwell equations are then solved using mode expansions, e.g. Fourier modes, which are well suited due to the periodic nature of the solution and known to be fast solution algorithms. Widely used packages like MIT Photonic Bands (MPB) [18] make use of this strategy. However, this method is also known to produce non-physical artefacts near discontinuities in the solution. Another approach is based on **Finite** **Element** **Methods** [21, 28, 9]. The basic idea is to define a weak formulation of the Maxwell equations and to approximate the solution using a piecewise polynomial basis, such that each basis function is non-zero only on a **finite** subset of the domain. This combines high flexibility with potentially high convergence rates.

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Organized by C.O.E.T, Akola & IWWA, Amravati Center. Available Online at www.ijpret.com 263 of **finite** size in the model. The term “**finite** **element** method”, was first introduced in solid mechanics applications, both used matrix displacement **methods** to solve plane stress problems with triangular and rectangular elements. Structural mechanics, continuum mechanics, flow and deformation are the main areas of application of **finite** **element** **methods**. Equilibrium problems often occur when the system does not vary with time of tensile test is represented in Figures [3 & 4], (Crisfield, 1986).

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Abstract: Combined with the characteristics of separable Hamiltonian systems and the **finite** **element** **methods** of ordinary differential equations, we prove that the composition of linear, quadratic, cubic **finite** **element** **methods** are symplectic integrator to separable Hamiltonian systems, i.e. the symplectic condition is preserved exactly, but the energy is only approximately conservative after compound. These conclusions are confirmed by our numerical experiments.

1. Introduction. Surface partial differential equations, i.e., partial differential equations on stationary or evolving surfaces, have become a flourishing mathematical field with numerous applications, e.g., in image processing [27], computer graphics [6], cell biology [22, 37], and porous media [35]. The numerical analysis of surface partial differential equations can be traced back to the pioneering paper of Dziuk [16] on the Laplace-Beltrami equation. Meanwhile there are various extensions to moving hypersurfaces such as, e.g., evolving surface **finite** **element** **methods** [17, 19] or trace **finite** **element** **methods** [39], and an abstract framework for parabolic equations on evolving Hilbert spaces [1, 2].

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The immersed **finite** **element** **methods** incorporate the **finite**-**element** discretization into the framework of the immersed-interface method for interface problems; and we therefore call them immersed-interface **finite**-**element** **methods** in this thesis. The ba- sic idea of an immersed-interface **finite**-**element** method is to incorporate the jump conditions in constructing basis functions. In a **finite**-difference immersed-interface method, the jump conditions are enforced through **finite**-difference equations on grid points near the interface. In immersed-interface **finite**-**element** **methods**, the jump conditions are enforced through the construction of special **finite**-**element** basis func- tions that satisfy the homogeneous interface conditions. Clearly, such basis functions depend on the interface location and the jump w. Some of the related work can be found in [12, 19, 46].

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are not inf-sup stable. Then, in the mid eighties the idea of stabilisation appeared in order to circumvent this restriction. Examples of stabilised **methods** are PSPG **methods**, introduced for shape-regular meshes in [HFB86], and extended to anisotropic meshes in [MPP03, Bla08]. Later on, different approaches have been proposed to stabilise this restriction, including Residual-Free Bubbles, or enriched **finite** **element** **methods** (see, e.g., [BBF93, ABV06]), and Local Projection Stabilised (LPS) **methods** (see [BB01]). Alternatively, and as an attempt to analyse some of these **methods** in a unified manner, the idea of minimal stabilisation was proposed in [BF01]. This approach consists of splitting the pressure space into the sum of a stable part and an unstable part. Then, a stabilising term is added to the formulation to control the unstable part of the pressure space, hence restoring stability. This approach provided a different interpretation of some older **methods**, e.g., [PS85], the local jumps stabilisation [KS92], and pressure projection [BDG06] (see [Bur08] for a unified presentation of this idea). In addition, this approach was recently used to design new inf-sup stable, and stabilised, **finite** **element** **methods** for the Stokes problem on anisotropic meshes in [ABW15] (modifying a decomposition given previously in [AC00]). This last work concerned for the pairs Q 2 k+1 × P 2 k − 1 , k ≥ 1, and then does not cover the lowest order case. For the latter case, and in the context of the Stokes problem, the local jump method from [KS92] was recently extended to anisotropic meshes without corner patches in [LS13], and to meshes containing corner patches in [BW15].

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12 November 1999, respectively. The investigations on the viaducts were carried out using both Geodetic and **Finite** **Element** **Methods** (FEM). Firstly, all the geodetic network stations selected for the project were checked because of the recent deformation in the area. Then, new control stations were placed between the piers of the viaducts. 28 object points were placed and measured on each pier to determine their displacements. In the second stage, the behaviours of the viaducts were modelled using the FEM, and the D¨uzce earthquake acceleration record was analysed to observe the response of the viaducts in a time history domain. The modelled displacement response of the viaducts was compared with the geodetic measurements in order to interpret the sensitivity of the design calculation of the engineering model. The pier displacements that were geodetically measured and calculated using FEM peak pier displacements showed an increase in the piers located closer to the surface rupture from the Izmit/Kocaeli and D¨uzce earthquakes. The agreement between the observed and modelled displacements decreases with the increase in the distance from the fault line. Since, near the fault trace the horizontal displacement field is discontinuous and large inelastic deformation is expected, the behaviour of the part of the structure located near the fault line cannot be

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Surface partial differential equations, i.e., partial differential equations on sta- tionary or evolving surfaces, have become a flourishing mathematical field with numerous applications, e.g., in image processing [26], computer graphics [6], cell biology [21, 34], and porous media [32]. The numerical analysis of surface partial differential equations can be traced back to the pioneering paper of Dziuk [15] on the Laplace-Beltrami equation. Meanwhile there are various extensions to moving hypersurfaces such as, e.g., evolving surface **finite** **element** **methods** [16, 17] or trace **finite** **element** **methods** [36], and an abstract framework for parabolic equations on evolving Hilbert spaces [1, 2].

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Before talking about **finite** **element** **methods** (FEMs), it is only right that one first talks of the partial differential equations (PDEs) they look to solve. PDEs have ex- isted as a mathematical model for all manner of physical phenomena for centuries, with equations describing how fluids flow, how heat transfers, and how sound waves propagate. Indeed capturing the essence of how the world works around us inher- ently requires complexity, meaning that in many cases the simpler ordinary differ- ential equation is not enough. Yet with this complexity requires an added effort to solve them, and often finding an analytical solution to all but the simplest PDEs is a difficult task, and at times an impossible one. Thus in modern times we typically look to numerical solutions and computers to solve PDEs, the most common meth- ods being the **finite** difference method, the **finite** volume method, and of course the **finite** **element** method.

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As FEM became widespread, its advantages and defects began to slowly sink in and further analy- sis became necessary. As its mathematical foundations were being uncovered, several mathematicians working in **finite** difference **methods** moved into FEM [101]. The mathematics of FEM started bloom- ing in the 1970’s and became a subject of worthy pursuit for budding mathematicians. In India, P. C. Das (IIT Kanpur) had started work on FEMs after his visit to Dundee for a conference in 1973 and a subsequent visit to Germany. P. K. Bhattacharyya (IIT Delhi) had started his work on FEMs after interactions with pioneers in FEMs like P.G. Ciarlet, O. Pironneau and M. Bernadou during his visits to France and O. Pironneau’s visits to India. The courses in mathematical theory of **finite** elements was intiated (thanks to suggestions of J. L. Lions to K. G. Ramanathan) in the 70’s with seminars by O. Pironneau followed by a full-fledged course [29] at IISc Bangalore in the year 1975 under the IISc-TIFR programme. This was followed by the TIFR lectures [74]. The Indo-French instructional conference and symposium held at the IISc-TIFR Centre in 1986 influenced and inspired A. K. Pani, who was then a graduate student of P. C. Das working in the area of FEMs.

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Recently, two effective numerical **methods** have been developed for analysing mechanical performance of heterogeneous materials [11-13]. The first is the so- called hybrid Trefftz FEM (or T-Trefftz method) [14, 15]. Unlike in the conventional FEM, the T-Trefftz method couples the advantages of conventional FEM [16-19] and BEM [20-22]. In contrast to the standard FEM, the T-Trefftz method is based on a hybrid method which includes the use of an independent auxiliary inter- **element** frame field defined on each **element** boundary and an independent internal field chosen so as to a prior satisfy the homogeneous governing differential equations by means of a suitable truncated T-complete function set of homogeneous solutions. Since 1970s, T-

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terms are deﬁned over the macroelements, and there is no error analysis or numerical validation of the method. All these works used the so-called bubble condensation procedure, i.e., eliminating the bubble function at the **element** level and writing the method as the Galerkin part, plus a term derived from the inﬂuence of the bubble functions on the formulation. A particular kind of bubble enrichment of the velocity space is the so-called residual-free bubble (RFB) method (cf. [7, 8, 12]), in which the bubble function is now the solution of a problem containing the residual of the continuous equation at the **element** level (see [9, 10, 32] for the a priori error analysis). This bubble part may be analytically condensed or numerically computed. In the latter case this procedure leads to the two-level ﬁnite **element** method.

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The starting point was the use of surface ﬁnite elements to compute solu- tions to the Poisson problem for the Laplace–Beltrami operator on a curved surface proposed and analysed in Dziuk (1988). Here an important con- cept is the use of triangulated surfaces on which ﬁnite **element** spaces are constructed and then used in variational formulations of surface PDEs us- ing surface gradients. This approach was extended by Dziuk and Elliott (2007b) to parabolic (including nonlinear and higher-order) equations on stationary surfaces. The evolving surface ﬁnite **element** method (ESFEM) was introduced by Dziuk and Elliott (2007a) in order to treat conservation laws on moving surfaces. The key idea is to use the Leibniz (or transport) formula for the time derivative of integrals over moving surfaces in order to derive weak and variational formulations. An interesting upshot is that the velocity and mean curvature of the surface which appear in certain for- mulations of the partial diﬀerential equation do not appear explicitly in variational formulations. This gives a tremendous advantage to numerical **methods** that exploit this, such as those of Dziuk and Elliott (2007a, 2010). Further numerical analysis of surface ﬁnite **element** **methods** may be found in Dziuk and Elliott (2012, 2013) and Dziuk, Lubich and Mansour (2012). Applications to complex physical and biological models may be found in Eilks and Elliott (2008), Elliott and Stinner (2010), Barreira, Elliott and Madzvamuse (2011) and Elliott, Stinner and Venkataraman (2012).

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In t his method the number of nodes per processor sti ll decreases as we move down t he levels, but t he fine grid relaxation sweeps may be performed in parallel with the coarse grid cor[r]

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