Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows
The focus of this monograph is the development of space-time adaptive methods to solve the convection/reaction dominated non-stationary semi-linear advection diffusion reaction (ADR) equations with internal/boundary layers in an accurate and efficient way. After introducing the ADR equations and discontinuous Galerkin discretization, robust residual-based a posteriori error estimators in space and time are derived. The elliptic reconstruction technique is then utilized to derive the a posteriori error bounds for the fully discrete system and to obtain optimal orders of convergence.As coupled surface and subsurface flow over large space and time scales is described by (ADR) equation the methods described in this book are of high importance in many areas of Geosciences including oil and gas recovery, groundwater contamination and sustainable use of groundwater resources, storing greenhouse gases or radioactive waste in the subsurface.
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Chapter 1 Introduction
Chapter 2 Discontinuous Galerkin Methods
Chapter 3 Elliptic Problems with Adaptivity
Chapter 4 Parabolic Problems with SpaceTime Adaptivity
Chapter 5 Conclusions and Outline
Appendix A Matlab Tutorial
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adaptive algorithm adaptive dG Adaptive Discontinuous Galerkin Adaptive mesh adaptive refinement basis polynomials BiCGStab bilinear form Cauchy–Schwarz inequality Chapter coef coercive compute condition number defined denote dG discretization dG methods dGAFEM diff Dirichlet boundary condition Discontinuous Galerkin Methods DoF's DoFs eeſ Eigenvector elliptic reconstruction technique error indicator Example finite element methods foliff formulation freact Fully Discrete GLSFEM ILU(S initial mesh interior penalty iterate Jacobian matrix Keč Laplacian matrix layers least squares Lemma linear reaction linear system Lipschitz continuous matrix reordering Neumann Boundary Condition Nloc Nodes non-linear reaction term non-stationary numerical solution obtain oscillations penalty parameter permutation permutation matrix Poisson problem posteriori error bounds posteriori error estimation preconditioner satisfying Schur complement semi-discrete semi-linear ADR equations SIAM SIPG solve space space-time adaptive sparse matrix sqrt stationary stiffness matrix SUPG-SC temporal estimator Theorem time-step triangle vector whlds