Hans-Görg Roos Martin Stynes Lutz Tobiska Robust Numerical Methods for Singularly Perturbed Differential Equations SPIN Springer s internal project number, if known Convection-Diffusion-Reaction and Flow Problems May 27, 2008 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo
Preface The analysis of singular perturbed differential equations began early in the twentieth century, when approximate solutions were constructed from asymptotic expansions. (Preliminary attempts appear in the nineteenth century see [vd94].) This technique has flourished since the mid-1960s and its principal ideas and methods are described in several textbooks; nevertheless, asymptotic expansions may be impossible to construct or may fail to simplify the given problem and then numerical approximations are often the only option. The systematic study of numerical methods for singular perturbation problems started somewhat later in the 1970s. From this time onwards the research frontier has steadily expanded, but the exposition of new developments in the analysis of these numerical methods has not received its due attention. The first textbook that concentrated on this analysis was [DMS80], which collected various results for ordinary differential equations. But after 1980 no further textbook appeared until 1996, when three books were published: Miller et al. [MOS96], which specializes in upwind finite difference methods on Shishkin meshes, Morton s book [Mor96], which is a general introduction to numerical methods for convection-diffusion problems with an emphasis on the cell-vertex finite volume method, and [RST96], the first edition of the present book. Nevertheless many methods and techniques that are important today, especially for partial differential equations, were developed after 1996. To give some examples, layer-adapted special meshes are frequently used, new stabilization techniques (such as discontinuous Galerkin methods and local subspace projections) are prominent, and there is a growing interest in the use of adaptive methods. Consequently contemporary researchers must comb the literature to gain an overview of current developments in this active area. In this second edition we retain the exposition of basic material that underpinned the first edition while extending its coverage to significant new numerical methods for singularly perturbed differential equations. Our purposes in writing this introductory book are twofold. First, we present a structured and comprehensive account of current ideas in the numerical analysis of singularly perturbed differential equations. Second, this
VI Preface important area has many open problems and we hope that our book will stimulate their investigation. Our choice of topics is inevitably personal and reflects our own main interests. We have learned a great deal about singularly perturbed problems from other researchers. We thank those colleagues who helped and influenced us; these include V.B. Andreev, A.E. Berger, P.A. Farrell, A. Felgenhauer, E.C. Gartland, Ch. Großmann, A.F. Hegarty, R.B. Kellogg, N. Kopteva, G. Lube, N. Madden, G. Matthies, J.J.H. Miller, K.W. Morton, F. Schieweck, G.I. Shishkin, E. Süli, and R. Vulanović; in particular Herbert Goering and Eugene O Riordan guided our initial steps in the area. Our research colleague T. Linß deserves additional thanks for providing many of the figures in this book. Our work was supported by the Deutsche Forschungsgemeinschaft and by the Boole Centre for Research in Informatics at the National University of Ireland, Cork. We are grateful to them, to the Mathematisches Forschungsinstitut in Oberwolfach for its hospitality, and to Springer-Verlag for its cooperation.
Contents Notation....................................................... XI Introduction................................................... 1 Part I Ordinary Differential Equations 1 The Analytical Behaviour of Solutions..................... 9 1.1 Linear Second-Order Problems Without Turning Points...... 11 1.1.1 Asymptotic Expansions............................ 12 1.1.2 The Green s Function and Stability Estimates......... 16 1.1.3 A Priori Estimates for Derivatives and Solution Decomposition.................................... 21 1.2 Linear Second-Order Turning-Point Problems............... 25 1.3 Quasilinear Problems.................................... 29 1.4 Linear Higher-Order Problems and Systems................. 35 1.4.1 Asymptotic Expansions for Higher-Order Problems.... 35 1.4.2 A Stability Result................................. 36 1.4.3 Systems of Ordinary Differential Equations........... 38 2 Numerical Methods for Second-Order Boundary Value Problems.................................................. 41 2.1 Finite Difference Methods on Equidistant Meshes............ 41 2.1.1 Classical Convergence Theory for Central Differencing. 41 2.1.2 Upwind Schemes.................................. 45 2.1.3 The Concept of Uniform Convergence................ 57 2.1.4 Uniformly Convergent Schemes of Higher Order....... 66 2.1.5 Linear Turning-Point Problems...................... 68 2.1.6 Some Nonlinear Problems.......................... 71 2.2 Finite Element Methods on Standard Meshes............... 76 2.2.1 Basic Results for Standard Finite Element Methods.... 76
VIII Contents 2.2.2 Upwind Finite Elements............................ 79 2.2.3 Stabilized Higher-Order Methods.................... 84 2.2.4 Variational Multiscale and Differentiated Residual Methods......................................... 95 2.2.5 Uniformly Convergent Finite Element Methods........ 104 2.3 Finite Volume Methods.................................. 114 2.4 Finite Difference Methods on Layer-adapted Grids........... 116 2.4.1 Graded Meshes................................... 119 2.4.2 Piecewise Equidistant Meshes....................... 127 2.5 Adaptive Strategies Based on Finite Differences............. 141 Part II Parabolic Initial-Boundary Value Problems in One Space Dimension 1 Introduction............................................... 155 2 Analytical Behaviour of Solutions.......................... 159 2.1 Existence, Uniqueness, Comparison Principle............... 159 2.2 Asymptotic Expansions and Bounds on Derivatives.......... 161 3 Finite Difference Methods................................. 169 3.1 First-Order Problems.................................... 169 3.1.1 Consistency...................................... 169 3.1.2 Stability......................................... 171 3.1.3 Convergence in L 2................................. 174 3.2 Convection-Diffusion Problems............................ 177 3.2.1 Consistency and Stability.......................... 178 3.2.2 Convergence...................................... 182 3.3 Polynomial Schemes..................................... 183 3.4 Uniformly Convergent Methods........................... 187 3.4.1 Exponential Fitting in Space........................ 188 3.4.2 Layer-Adapted Tensor-Product Meshes............... 189 3.4.3 Reaction-Diffusion Problems........................ 191 4 Finite Element Methods................................... 195 4.1 Space-Based Methods.................................... 196 4.1.1 Polynomial Upwinding............................. 197 4.1.2 Uniformly Convergent Schemes...................... 199 4.1.3 Local Error Estimates............................. 203 4.2 Subcharacteristic-Based Methods.......................... 205 4.2.1 SDFEM in Space-Time............................. 206 4.2.2 Explicit Galerkin Methods.......................... 211 4.2.3 Eulerian-Lagrangian Methods....................... 217
Contents IX 5 Two Adaptive Methods.................................... 223 5.1 Streamline Diffusion Methods............................. 223 5.2 Moving Mesh Methods (r-refinement)...................... 225 Part III Elliptic and Parabolic Problems in Several Space Dimensions 1 Analytical Behaviour of Solutions.......................... 235 1.1 Classical and Weak Solutions............................. 235 1.2 The Reduced Problem................................... 238 1.3 Asymptotic Expansions and Boundary Layers............... 243 1.4 A Priori Estimates and Solution Decomposition............. 247 2 Finite Difference Methods................................. 259 2.1 Finite Difference Methods on Standard Meshes.............. 259 2.1.1 Exponential Boundary Layers....................... 259 2.1.2 Parabolic Boundary Layers......................... 266 2.2 Layer-Adapted Meshes................................... 268 2.2.1 Exponential Boundary Layers....................... 268 2.2.2 Parabolic Layers.................................. 274 3 Finite Element Methods................................... 277 3.1 Inverse-Monotonicity-Preserving Methods Based on Finite Volume Ideas........................................... 278 3.2 Residual-Based Stabilizations............................. 302 3.2.1 Streamline Diffusion Finite Element Method (SDFEM). 302 3.2.2 Galerkin Least Squares Finite Element Method (GLSFEM)....................................... 327 3.2.3 Residual-Free Bubbles............................. 333 3.3 Adding Symmetric Stabilizing Terms....................... 338 3.3.1 Local Projection Stabilization....................... 338 3.3.2 Continuous Interior Penalty Stabilization............. 352 3.4 The Discontinuous Galerkin Finite Element Method......... 363 3.4.1 The Primal Formulation for a Reaction-Diffusion Problem.......................................... 363 3.4.2 A First-Order Hyperbolic Problem.................. 368 3.4.3 dgfem Error Analysis for Convection-Diffusion Problems......................................... 371 3.5 Uniformly Convergent Methods........................... 376 3.5.1 Operator-Fitted Methods........................... 377 3.5.2 Layer-Adapted Meshes............................. 381 3.6 Adaptive Methods....................................... 407 3.6.1 Adaptive Finite Element Methods for Non-Singularly Perturbed Elliptic Problems: an Introduction......... 407
X Contents 3.6.2 Robust and Semi-Robust Residual Type Error Estimators....................................... 414 3.6.3 A Variant of the DWR Method for Streamline Diffusion 421 4 Time-Dependent Problems................................ 427 4.1 Analytical Behaviour of Solutions.......................... 428 4.2 Finite Difference Methods................................ 429 4.3 Finite Element Methods.................................. 434 Part IV The Incompressible Navier-Stokes Equations 1 Existence and Uniqueness Results......................... 449 2 Upwind Finite Element Method........................... 453 3 Higher-Order Methods of Streamline Diffusion Type...... 465 3.1 The Oseen Problem...................................... 466 3.2 The Navier-Stokes Problem............................... 476 4 Local Projection Stabilization for Equal-Order Interpolation485 4.1 Local Projection Stabilization in an Abstract Setting......... 486 4.2 Convergence Analysis.................................... 488 4.2.1 The Special Interpolant............................ 488 4.2.2 Stability......................................... 489 4.2.3 Consistency Error................................. 491 4.2.4 A priori Error Estimate............................ 492 4.3 Local Projection onto Coarse-Mesh Spaces.................. 498 4.3.1 Simplices......................................... 498 4.3.2 Quadrilaterals and Hexahedra....................... 499 4.4 Schemes Based on Enrichment of Approximation Spaces...... 501 4.4.1 Simplices......................................... 502 4.4.2 Quadrilaterals and Hexahedra....................... 502 4.5 Relationship to Subgrid Modelling......................... 504 4.5.1 Two-Level Approach with Piecewise Linear Elements.. 505 4.5.2 Enriched Piecewise Linear Elements................. 507 4.5.3 Spectral Equivalence of the Stabilizing Terms on Simplices......................................... 508 5 Local Projection Method for Inf-Sup Stable Elements..... 511 5.1 Discretization by Inf-Sup Stable Elements.................. 512 5.2 Stability and Consistency................................. 514 5.3 Convergence............................................ 516 5.3.1 Methods of Order r in the Case σ > 0............... 517 5.3.2 Methods of Order r in the Case σ 0............... 522 5.3.3 Methods of Order r + 1/2......................... 526
Notation XI 6 Mass Conservation for Coupled Flow-Transport Problems. 529 6.1 A Model Problem....................................... 529 6.2 Continuous and Discrete Mass Conservation................ 530 6.3 Approximated Incompressible Flows....................... 532 6.4 Mass-Conservative Methods............................... 534 6.4.1 Higher-Order Flow Approximation................... 534 6.4.2 Post-Processing of the Discrete Velocity.............. 536 6.4.3 Scott-Vogelius Elements............................ 542 7 Adaptive Error Control.................................... 545 References..................................................... 551 Index.......................................................... 599