Contents. I The Basic Framework for Stationary Problems 1

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1 page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs Laplace s equation; elliptic BVPs Physical experiments modeled by Laplace s equation Other elliptic BVPs The equations of isotropic elasticity General linear elasticity Exercises for Chapter The weak form of a BVP Review of vector calculus The divergence theorem Green s identity Other forms of the divergence theorem and Green s identity The weak form of a BVP Minimization of energy Relaxing the PDE A few details about Sobolev spaces The weak form for other boundary conditions and PDEs Neumann conditions and the weak form Mixed boundary conditions Inhomogeneous boundary conditions Other elliptic BVPs Existence and uniqueness theory for the weak form of a BVP Vector spaces and inner products Hilbert spaces Linear functionals The Riesz representation theorem Variational problems and the Riesz representation theorem 42 vii

2 page v viii 2.5 Examples of ellipticity The model problem The equations of isotropic elasticity Variational formulation of nonsymmetric problems Exercises for Chapter The Galerkin method The projection theorem The Galerkin method for a variational problem Another interpretation of the Galerkin method The Galerkin method for a nonsymmetric problem Exercises for Chapter Piecewise polynomials and the finite element method Piecewise linear functions defined on a triangular mesh Using piecewise linear functions in Galerkin s method The sparsity of the stiffness matrix Quadratic Lagrange triangles Continuous piecewise quadratic functions The finite element method with quadratic Lagrange triangles Cubic Lagrange triangles Continuous piecewise cubic functions The finite element method with cubic Lagrange triangles Lagrange triangles of arbitrary degree Hierarchical bases for finite element spaces Other finite elements: Rectangles and quadrilaterals Rectangular elements General quadrilaterals Using a reference triangle in finite element calculations Isoparametric finite element methods Isoparametric quadratic triangles Isoparametric triangles of higher degree Exercises for Chapter Convergence of the finite element method Approximating smooth functions by continuous piecewise linear functions The standard refinement of a triangulation Nondegenerate families of triangulations Approximation by piecewise linear functions Approximation by higher-order piecewise polynomials Convergence in the energy norm Convergence in the L 2 -norm Variational crimes Numerical integration...118

3 page ix ix Outline of the analysis of the effect of quadrature Isoparametric finite elements Exercises for Chapter II Data Structures and Implementation The mesh data structure Programming the finite element method Assembling the stiffness matrix Computing the load vector The mesh data structure The list of nodes The list of edges The list of elements The list of free boundary edges Other fields in the mesh data structure The MATLAB implementation Generating a mesh by refinement Generating a mesh from a triangle-node list Assessing the quality of a triangulation Viewing a mesh Handling a domain with a curved boundary Viewing a piecewise linear function MATLAB functions A summary of the notation Exercises for Chapter Programming the finite element method: Linear Lagrange triangles Quadrature Gaussian quadrature Evaluating the standard basis functions on a triangle Quadrature over a square Assembling the stiffness matrix Computing the load vector Inhomogeneous Dirichlet conditions Inhomogeneous Neumann conditions Examples Homogeneous boundary conditions Inhomogeneous boundary conditions A more realistic example The MATLAB implementation MATLAB functions Exercises for Chapter

4 pagex x 8 Lagrange triangles of arbitrary degree Quadrature for higher-order elements Assembling the stiffness matrix and load vector Implementing the isoparametric method Placement of nodes in the isoparametric method Examples The MATLAB implementation version version Exercises for Chapter The finite element method for general BVPs Scalar BVPs An example Isotropic elasticity Mesh locking The MATLAB implementation Exercises for Chapter III Solving the Finite Element Equations Direct solution of sparse linear systems The Cholesky factorization for positive definite matrices The Cholesky factorization for dense matrices The Cholesky factorization for banded matrices Factoring general sparse matrices Exercises for Chapter Iterative methods: Conjugate gradients The CG method The CG algorithm Convergence of the CG algorithm Hierarchical bases for finite element spaces Hierarchical bases for linear Lagrange triangles Relationship between the stiffness matrices in nodal and hierarchical bases The hierarchical basis CG method The preconditioned CG method Alternate derivation of PCG Preconditioners The pure Neumann problem The MATLAB implementation MATLAB functions Exercises for Chapter

5 page x xi 12 The classical stationary iterations Stationary iterations Matrix norms Convergence of stationary iterations The classical iterations Jacobi iteration Gauss Seidel iteration SOR iteration Symmetric SOR CG with SSOR preconditioning The MATLAB implementation MATLAB functions Exercises for Chapter The multigrid method Stationary iterations as smoothers The stiffness matrix for the model problem Fourier modes and the spectral decomposition of K Jacobi iteration Weighted Jacobi iteration The coarse grid correction algorithm Projecting the equation onto a coarser mesh The projected equation and the Galerkin idea The two-grid multigrid algorithm The multigrid V-cycle W-cycles and µ-cycles Full multigrid Discretization, algebraic, and total errors The MATLAB implementation MATLAB functions Exercises for Chapter IV Adaptive Methods Adaptive mesh generation Algorithms for local mesh refinement Algorithms based on the standard refinement Algorithms based on bisection Selecting triangles for local refinement A complete adaptive algorithm The MATLAB implementation MATLAB functions Exercises for Chapter

6 page x xii 15 Error estimators and indicators An explicit error indicator based on estimating the curvature of the solution An explicit error indicator based on the residual The element residual error estimator Some final examples A discontinuous coefficient A reentrant corner Transition from Dirichlet to Neumann conditions The MATLAB implementation MATLAB functions Exercises for Chapter Bibliography 353 Index 357

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 20: Sparse Linear Systems; Direct Methods vs. Iterative Methods Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 26