Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications. In Honor of Professor Raytcho Lazarov’s 40 Years of Research in Computational Methods and Applied Mathematics

Description

Preface Contents Improving Conservation for First-Order System Least-Squares Finite-Element Methods 1 Introduction 2 First-Order System Least-Squares 2.1 Sample Problem and Loss of Conservation 3 Constrained FOSLS 3.1 ``Galerkin Constraint'' 3.2 ``Least-Squares Constraint'' 3.3 Solvers 3.3.1 Construction of B 3.4 Numerical Results 3.5 Discussion 4 Locally Constrained FOSLS Correction 4.1 Overlapping Schwarz Corrections 4.2 Numerical Results 5 Conclusions References Multiscale Coarsening for Linear Elasticity by Energy Minimization 1 Introduction 2 Governing Equations and Their Discretization 2.1 The Equations of Linear Elasticity 2.2 Weak Formulation 2.3 Finite Element Discretization 3 The Two-Level Method 3.1 Two-Level Additive Schwarz 3.2 Fine and Coarse Triangulation 3.3 Abstract Multiscale Coarse Space 4 Energy Minimization for the Elasticity System 4.1 The Energy-Minimizing Coarse Space 4.2 Properties of the Energy-Minimizing Coarse Space 4.3 The Interpolation Operator 5 Numerical Experiments 5.1 Coarse Space Robustness 5.2 Coarse Space Approximation 6 Discussion References Preconditioners for Some Matrices of Two-by-Two Block Form, with Applications, I 1 Introduction 2 The Preconditioner and Its Implementation 3 Condition Number Bounds 3.1 A Is Symmetric and Positive Semidefinite 3.2 A Is Symmetric and Positive Definite 4 Distributed Optimal Control of Elliptic and Oseen Equations 4.1 An Elliptic State Equation 4.2 Distributed Optimal Control of the Oseen Problem References A Multigrid Algorithm for an Elliptic Problem with a Perturbed Boundary Condition 1 Introduction 2 Preliminaries 3 The Multigrid Algorithm 4 Multigrid Analysis 5 Numerical Results References Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs 1 Introduction 2 Unsmoothed Aggregation AMG 3 The Setup Phase 3.1 A Maximal Independent Set Algorithm 3.2 Parallel Graph Aggregation Algorithm 3.3 Aggregation Quality Improvements 4 Solve Phase 4.1 Parallel Smoother 4.2 Prolongation and Restriction 4.3 K-Cycle 4.4 Sparse Matrix-Vector Multiplication on GPUs 5 Numerical Tests 5.1 Numerical Tests for PAA 5.2 Numerical Tests for GPU Implementation References Aspects of Guaranteed Error Control in CPDEs 1 Introduction 2 Review of Guaranteed Energy Norm Error Control 2.1 Notation 2.2 Equilibration Error Estimators 2.3 Poisson Model Problem with Big Oscillations 3 Guaranteed Error Control for CR-NCFEM 3.1 Main Result 3.2 Guaranteed Upper Bounds for ||Res|||* 3.3 Guaranteed Upper Bounds for ||||ResNC|||| 3.4 Numerical Experiment with Big Oscillations 4 Guaranteed Error Control for Curved Boundaries 5 Guaranteed Goal-Oriented Error Estimation 5.1 Reduction to L2 Functionals 5.2 Guaranteed Bounds for Goal Functionals 5.3 Benchmark Example References A Finite Volume Element Method for a Nonlinear Parabolic Problem 1 Introduction 2 Preliminaries 3 Error Estimates for the Backward Euler Method 4 Existence of the Backward Euler Approximation 5 A Linearized Fully Discrete Scheme 6 Numerical Examples References Multidimensional Sensitivity Analysis of Large-Scale Mathematical Models 1 Introduction 2 Problem Setting 2.1 Modeling and Sensitivity 3 Complexity in Classes of Algorithms 4 The Algorithms 4.1 ΛΠτ Sobol Sequences 4.2 The Monte Carlo Algorithms Based on Modified Sobol Sequences: MCA-MSS 5 Case Study: Variance-Based Sensitivity Analysis of the Unified Danish Eulerian Model 6 Numerical Results and Discussion 7 Conclusions References Structures and Waves in a Nonlinear Heat-Conducting Medium 1 Introduction 2 The Radially Symmetric Case, the Main Notions 2.1 The Basic Blow-Up Regimes 2.1.1 HS-Evolution, Total Blow-Up, 1