Partial Differential Equations and the Finite Element Method

Transcription

1 Partial Differential Equations and the Finite Element Method Pavel Solin The University of Texas at El Paso Academy of Sciences ofthe Czech Republic iwiley- INTERSCIENCE A JOHN WILEY & SONS, INC, PUBLICATION

2 CONTENTS List of Figures List of Tables Preface Acknowledgments xv xxi xxiii xxv 1 Partial Differential Equations Selected general properties Classification and examples Hadamard's well-posedness General existence and uniqueness results Exercises Second-order elliptic problems Weak formulation of a model problem Bilinear forms, energy norm, and energetic inner product The Lax-Milgram lemma Unique solvability of the model problem Nonhomogeneous Dirichlet boundary conditions Neumann boundary conditions Newton (Robin) boundary conditions Combining essential and natural boundary conditions 23

3 VIII CONTENTS Energy of elliptic problems Maximum principles and well-posedness Exercises Second-order parabolic problems Initial and boundary conditions Weak formulation Existence and uniqueness of Solution Exercises Second-order hyperbolic problems Initial and boundary conditions Weak formulation and unique solvability The wave equation Exercises First-order hyperbolic problems Conservation laws Characteristics Exact Solution to linear first-order Systems Riemann problem Nonlinearfluxand shock formation Exercises 44 2 Continuous Elements for 1D Problems The general framework The Galerkin method Orthogonality of error and Cea's lemma Convergence of the Galerkin method Ritz method for Symmetrie problems Exercises Lowest-order elements Model problem Finite-dimensional subspace V n cv Piecewise-affine basis funetions The system of linear algebraic equations Element-by-element assembling procedure Refinement and convergence Exercises Higher-order numerical quadrature Gaussian quadrature rules Selected quadrature constants Adaptive quadrature Exercises Higher-order elements 66

4 CONTENTS IX Motivation problem Affine concept: reference domain and reference maps Transformation of weak forms to the reference domain Higher-order Lagrange nodal shape functions Chebyshev and Gauss-Lobatto nodal points Higher-order Lobatto hierarchic shape functions Constructing basis of the Space Vh tp Data structures Assembling algorithm Exercises The sparse stiffness matrix Compressed sparse row (CSR) data format Condition number Conditioning of shape functions Stiffness matrix for the Lobatto shape functions Exercises Implementing nonhomogeneous boundary conditions Dirichlet boundary conditions Combination of essential and natural conditions Exercises Interpolation onfiniteelements The Hubert space setting Best interpolant Projection-based interpolant Nodal interpolant Exercises 102 General Concept of Nodal Elements The nodalfinitedement Unisolvency and nodal basis Checking unisolvency Example: Iowest-orderQ 1 - and P 1 -elements Q^element P^element Invertibility of the quadrilateral reference map XK Interpolation on nodal elements Local nodal interpolant Global interpolant and conformity Conformity to the Sobolev space H Equivalence of nodal elements Exercises 122

5 X CONTENTS 4 Continuous Elements for 2D Problems Lowest-order elements Model problem and its weak formulation Approximations and variational crimes Basis of the space Vft ip Transformation of weak forms to the reference domain Simplified evaluation of stiffness integrals Connectivity arrays Assembling algorithm for Q 1 /P 1 -elements Lagrange interpolation on Q 1 /P 1 -meshes Exercises Higher-order numerical quadrature in 2D Gaussian quadrature on quads Gaussian quadrature on triangles Higher-order nodal elements Product Gauss-Lobatto points Lagrange-Gauss-Lobatto <3 p ' r -elements Lagrange interpolation and the Lebesgue constant The Fekete points Lagrange-Fekete P p -elements Basis of the space Vh ip Data structures Connectivity arrays Assembling algorithm for Q p /P p -elements Lagrange interpolation on Q p /P p -meshes Exercises Transient Problems and ODE Solvers Methodoflines Model problem Weak formulation The ODE System Construction of the initial vector Autonomous Systems and phase flow Selected time integration schemes One-step methods, consistency and convergence Explicit and implicit Euler methods Stiffness Explicit higher-order RK schemes Embedded RK methods and adaptivity General (implicit) RK schemes 184

6 CONTENTS Xi 5.3 Introduction to stability Autonomization of RK methods Stability of linear autonomous Systems Stability functions and stability domains Stability functions for general RK methods Maximum consistency order of IRK methods yl-stability and L-stability Higher-order IRK methods Collocation methods Gauss and Radau IRK methods Solution of nonlinear Systems Exercises Beam and Plate Bending Problems Bending of elastic beams Euler-Bernoulli model Boundary conditions Weak formulation Existence and uniqueness of Solution Lowest-order Hermite elements in 1D Model problem Cubic Hermite elements Higher-order Hermite elements in 1D Nodal higher-order elements Hierarchie higher-order elements Conditioning of shape functions Basis of the space Vh ip Transformation of weak forms to the reference domain Connectivity arrays Assembling algorithm Interpolation on Hermite elements Hermite elements in 2D Lowest-order elements Higher-order Hermite-Fekete elements Design of basis functions Global nodal interpolant and conformity Bending of elastic plates Reissner-Mindlin (thick) plate model Kirchhoff (thin) plate model Boundary conditions Weak formulation and unique solvability Babuska's paradox of thin plates 254

7 XII CONTENTS 6.6 Discretization by i7 2 -conforming elements Lowest-order (quintic) Argyris element, unisolvency Local interpolant, conformity Nodal shape functions on the reference domain Transformation to reference domains Design of basis functions Higher-order nodal Argyris-Fekete elements Exercises 266 Equations of Electromagnetics Electromagnetic field and its basic characteristics Integration along smooth curves Maxwell's equations in integral form Maxwell's equations in differential form Constitutive relations and the equation of continuity Media and their characteristics Conductors and dielectrics Magnetic materials Conditions on interfaces Potentials Scalar electric potential Scalar magnetic potential Vector potential and gauge transformations Potential formulation of Maxwell's equations Other wave equations Equations for thefieldvectors Equation for the electric field Equation for the magnetic field Interface and boundary conditions Time-harmonic Maxwell's equations Helmholtz equation Time-harmonic Maxwell's equations Normalization Model problem Weak formulation Existence and uniqueness of Solution Edge elements Conformity requirements of the space if(curl) Lowest-order (Whitney) edge elements Higher-order edge elements of N6delec Transformation of weak forms to the reference domain Interpolation on edge elements 316

8 CONTENTS XIII Conformity of edge elements to the space ü(curl) Exercises 318 Appendix A: Basics of Functional Analysis 319 A.l Linear Spaces 320 A. 1.1 Real and complex linear space 320 A. 1.2 Checking whether a set is a linear space 321 A. 1.3 Intersection and union of subspaces 323 A. 1.4 Linear combination and linear span 326 A. 1.5 Sum and direct sum of subspaces 327 A.1.6 Linear independence, basis, and dimension 328 A.1.7 Linear Operator, null space, ränge 332 A. 1.8 Composed Operators and change of basis 337 A.1.9 Determinants, eigenvalues, and eigenvectors 339 A.l.10 Hermitian, Symmetrie, and diagonalizable matrices 341 A Linear forms, dual space, and dual basis 343 A.l. 12 Exercises 345 A.2 Normed Spaces 348 A.2.1 Norm and seminorm 348 A.2.2 Convergence and limit 352 A.2.3 Open and closed sets 355 A.2.4 Continuity of Operators 357 A.2.5 Operator norm and C(U, V) as a normed space 361 A.2.6 Equivalence of norms 363 A.2.7 Banach Spaces 366 A.2.8 Banach fixed point theorem 371 A.2.9 Lebesgue integral and L p -spaces 375 A.2.10 Basic inequalities in ZAspaces 380 A.2.11 Density of smooth funetions in L p -spaces 384 A.2.12 Exercises 386 A.3 Inner produet Spaces 389 A.3.1 Inner produet 389 A.3.2 Hubert Spaces 394 A.3.3 Generalized angle and orthogonality 395 A.3.4 Generalized Fourier series 399 A.3.5 Projections and orthogonal projeetions 401 A.3.6 Representation of linear forms (Riesz) 405 A.3.7 Compactness, compact Operators, and the Fredholm alternative 407 A.3.8 Weak convergence 408 A.3.9 Exercises 409 A.4 Sobolev Spaces 412 A.4.1 Domain boundary and its regularity 412

9 XiV CONTENTS A.4.2 Distributions and weak derivatives 414 A.4.3 Spaces W k ' p and H k 418 A.4.4 Discontinuity ofi^-functions in R d, d> A.4.5 Poincare-Friedrichs' inequality 421 A.4.6 Embeddings of Sobolev Spaces 422 A.4.7 Traces of W fc - p -functions 424 A.4.8 Generalized Integration by parts formulae 425 A.4.9 Exercises 426 Appendix B: Software and Examples 427 B.l Sparse Matrix Solvers 427 B. 1.1 The smatrix Utility 428 B.1.2 An example application 430 B.1.3 Interfacing with PETSc 433 B. 1.4 Interfacing with Trilinos 436 B. 1.5 Interfacing with UMFPACK 439 B.2 The High-Performance Modular Finite Element System HERMES 439 B.2.1 Modular structure of HERMES 440 B.2.2 The elliptic module 441 B.2.3 The Maxwell's module 442 B.2.4 Example 1: L-shape domain problem 444 B.2.5 Example 2: Insulator problem 448 B.2.6 Example 3: Sphere-cone problem 451 B.2.7 Example 4: Electrostatic micromotor problem 455 B.2.8 Example 5: Diffraction problem 458 References 461 Index 468