Discontinuous Galerkin and finite element methods
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Weihua Deng
Abstract
This chapter is a short essay on discontinuous Galerkin methods for fractional (convection-) diffusion equations in one and two dimensions. The method is based on the local discontinuous Galerkin methods for the classical parabolic equation, i. e., decomposing the high-order derivative and rewriting the equation into a first-order system. Depending on the properties of fractional operators, we decompose it into several first-order derivatives and one fractional integral. Then we propose the corresponding numerical schemes and discuss their stability and convergence. Some algorithms in two dimensions are provided.
Abstract
This chapter is a short essay on discontinuous Galerkin methods for fractional (convection-) diffusion equations in one and two dimensions. The method is based on the local discontinuous Galerkin methods for the classical parabolic equation, i. e., decomposing the high-order derivative and rewriting the equation into a first-order system. Depending on the properties of fractional operators, we decompose it into several first-order derivatives and one fractional integral. Then we propose the corresponding numerical schemes and discuss their stability and convergence. Some algorithms in two dimensions are provided.
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations 1
- Time-fractional derivatives 23
- High-order finite difference methods for fractional partial differential equations 49
- Spectral methods for some kinds of fractional differential equations 101
- Spectral methods for fractional differential equations using generalized Jacobi functions 127
- Spectral and spectral element methods for fractional advection–diffusion–reaction equations 157
- Discontinuous Galerkin and finite element methods 185
- Numerical methods for time-space fractional partial differential equations 209
- Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian 249
- Particle tracking solutions of vector fractional differential equations: A review 275
- Singularities 287
- Fast numerical methods for space-fractional partial differential equations 307
- Fast methods for the computation of the Mittag-Leffler function 329
- Index 347
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations 1
- Time-fractional derivatives 23
- High-order finite difference methods for fractional partial differential equations 49
- Spectral methods for some kinds of fractional differential equations 101
- Spectral methods for fractional differential equations using generalized Jacobi functions 127
- Spectral and spectral element methods for fractional advection–diffusion–reaction equations 157
- Discontinuous Galerkin and finite element methods 185
- Numerical methods for time-space fractional partial differential equations 209
- Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian 249
- Particle tracking solutions of vector fractional differential equations: A review 275
- Singularities 287
- Fast numerical methods for space-fractional partial differential equations 307
- Fast methods for the computation of the Mittag-Leffler function 329
- Index 347
