In this thesis we develop 2D parallel unstructured mesh adaptation methods for the solution of partial differential equations (PDEs) by the finite element method (FEM). Additionally, we develop a novel block preconditioner for the iterative solution of the linear systems arising from the finite element discretisation of the Föppl-von Kàrmàn equations. Two of the problems arising in the numerical solution of PDEs by FEM are the memory constraints that limit the solution of large problems, and the inefficiency of solving the associated linear systems by direct or iterative solvers. We initially focus on mesh adaptation, which is a memory demanding task of the FEM. The size of the problem increases by adding more elements and nodes to the mesh during mesh refinement. In problems involving a large number of elements, the problem size is limited by the memory available on a single processor. In order to be able to work with large problems, we use a domain decomposition approach to distribute the problem over multiple processors. One of the main objectives of this thesis is the development of 2D parallel unstructured mesh adaptation methods for the solution of PDEs by the FEM in a variety of problems; including domains with curved boundaries, holes and internal boundaries. Our newly developed methods are implemented in the software library oomph-lib, an open-source object oriented multi-physics software library implementing the FEM. We validate and demonstrate their utility in a set of increasingly complex problems ranging from scalar PDEs to fully coupled multi-physics problems. Having implemented and validated the infrastructure which facilitates the finite-element-based discretisation of PDEs in a distributed environment, we shift our focus to the second problem concerning this thesis and one of the major challenges in the computational solution of PDEs: the solution of the large linear systems arising from their discretisation. For sufficiently large problems, the solution of their associated linear system by direct solvers becomes impossible or inefficient, typically because of memory and time constraints. We therefore focus on preconditioned Krylov subspace methods whose efficiency depends crucially on the provision of a good preconditioner. These preconditioners are invariably problem dependent. We illustrate their application and development in the solution of two elasticity problems which give rise to relatively large problems. First we consider the solution of a linear elasticity problem and compute the stress distribution near a crack tip where strong local mesh refinement is required. We then consider the deformation of thin plates which are described by the nonlinear Föppl-von Kàrmàn equations. A key contribution of this work is the development of a novel block preconditioner for the iterative solution of these equations, we present the development of the preconditioner and demonstrate its practical performance.

In this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.

In this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.

Several advancements related to the solution of 3D time-harmonic wave equations are presented, especially in the context of a parallel moving-PML sweeping preconditioner for problems without large-scale resonances. The main contribution of this dissertation is the introduction of an efficient parallel sweeping preconditioner and its subsequent application to several challenging velocity models. For instance, 3D seismic problems approaching a billion degrees of freedom have been solved in just a few minutes using several thousand processors. The setup and application costs of the sequential algorithm were also respectively refined to O(γ^2 N^(4/3)) and O(γ N log N), where N denotes the total number of degrees of freedom in the 3D volume and γ(ω) denotes the modestly frequency-dependent number of grid points per Perfectly Matched Layer discretization. Furthermore, high-performance parallel algorithms are proposed for performing multifrontal triangular solves with many right-hand sides, and a custom compression scheme is introduced which builds upon the translation invariance of free-space Green’s functions in order to justify the replacement of each dense matrix within a certain modified multifrontal method with the sum of a small number of Kronecker products. For the sake of reproducibility, every algorithm exercised within this dissertation is made available as part of the open source packages Clique and Parallel Sweeping Preconditioner (PSP). / text

Two solution strategies for large, sparse, and structured algebraic systems of equations are considered. The first strategy is to construct efficient preconditioners for iterative solvers. The second is to reduce the sparse algebraic system to a smaller, dense system of equations, which are called the boundary summation equations. The proposed preconditioners perform well when applied to equations that are discretizations of linear first order partial differential equations. Analysis shows that also very simple iterative methods converge in a number of iterations that is independent of the number of unknowns, if our preconditioners are applied to certain scalar model problems. Numerical experiments indicate that this property holds also for more complicated cases, and a flow problem modeled by the nonlinear Euler equations is treated successfully. The reduction process is applicable to a large class of difference equations. There is no approximation involved in the reduction, so the solution of the original algebraic equations is determined exactly if the reduced system is solved exactly. The reduced system is well suited for iterative solution, especially if the original system of equations is a discretization of a first order differential equation. The technique is used for several problems, ranging from scalar model problems to a semi-implicit discretization of the compressible Navier-Stokes equations. Both strategies use the concept of fundamental solutions, either of differential or difference operators. An algorithm for computing fundamental solutions of difference operators is also presented.

After some remarks on the parallel implementation of the Finite Element package FEAP, our realisation of the parallel CG-algorithm is sketched. From a technical point of view, a hierarchical preconditioner with and without additional global crosspoint preconditioning is presented. The numerical properties of this preconditioners are discussed and compared to a Schur-complement-preconditioning, using a wide range of data from computations on technical and academic examples from elasticity.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the p-version finite element matrixes. Also DD preconditioning is considered.

Numerical techniques for linear systems arising from discretization of partial differential equations are nowadays essential for understanding the physical world. Among these techniques, iterative methods and the accompanying preconditioning techniques have become increasingly popular due to their great potential on large scale computation. In this work, we present preconditioning techniques for linear systems built with tensor product basis functions. Efficient algorithms are designed for various problems by exploiting the Kronecker product structure in the matrices, inherited from tensor product basis functions. Specifically, we design preconditioners for mass matrices to remove the complexity from the basis functions used in isogeometric analysis, obtaining numerical performance independent of mesh size, polynomial order and continuity order; we also present a compound iteration preconditioner for stiffness matrices in two dimensions, obtaining fast convergence speed; lastly, for the Helmholtz problem, we present a strategy to `hide' its indefiniteness from Krylov subspace methods by eliminating the part of initial error that corresponds to those negative generalized eigenvalues. For all three cases, the Kronecker product structure in the matrices is exploited to achieve high computational efficiency.

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The availability of high performance computing clusters has allowed scientists and engineers to study more challenging problems. However, new algorithms need to be developed to take advantage of the new computer architecture (in particular, distributed memory clusters). Since the solution of linear systems still demands most of the computational effort in many problems (such as the approximation of partial differential equation models) iterative methods and, in particular, efficient preconditioners need to be developed. In this study, we consider application of incomplete LU (ILU) preconditioners for finite element models to partial differential equations. Since finite elements lead to large, sparse systems, reordering the node numbers can have a substantial influence on the effectiveness of these preconditioners. We study two implementations of the ILU preconditioner: a stucturebased method and a threshold-based method. The main emphasis of the thesis is to test a variety of breadth-first ordering strategies on the convergence properties of the preconditioned systems. These include conventional Cuthill-McKee (CM) and Reverse Cuthill-McKee (RCM) orderings as well as strategies related to the physical distance between nodes and post-processing methods based on relative sizes of associated matrix entries. Although the success of these methods were problem dependent, a number of tendencies emerged from which we could make recommendations. Finally, we perform a preliminary study of the multi-processor case and observe the importance of partitioning quality and the parallel ILU reordering strategy. / Master of Science