hp-Finite Element Methods for Singular Perturbations

1. Front Matter

   Pages N2-XIV

   PDF

2. 1. Introduction

   • Jens M. Melenk

   Pages 1-20

3. 2. hp-FEM for Reaction Diffusion Problems: Principal Results

   • Jens M. Melenk

   Pages 23-72

4. 3. hp Approximation

   • Jens M. Melenk

   Pages 73-138

5. 4. The Countably Normed Spaces

   • Jens M. Melenk

   Pages 141-168

6. 5. Regularity Theory in Countably Normed Spaces

   • Jens M. Melenk

   Pages 169-224

7. 6. Exponentially Weighted Countably Normed Spaces

   • Jens M. Melenk

   Pages 227-254

8. 7. Regularity through Asymptotic Expansions

   • Jens M. Melenk

   Pages 255-295

9. Appendix

   • Jens M. Melenk

   Pages 297-310

10. References

    • Jens M. Melenk

    Pages 311-316

11. Index

    • Jens M. Melenk

    Pages 317-318

12. Back Matter

    Pages 319-323

    PDF

Many partial differential equations arising in practice are parameter-dependent problems that are of singularly perturbed type. Prominent examples include plate and shell models for small thickness in solid mechanics, convection-diffusion problems in fluid mechanics, and equations arising in semi-conductor device modelling. Common features of these problems are layers and, in the case of non-smooth geometries, corner singularities. Mesh design principles for the efficient approximation of both features by the hp-version of the finite element method (hp-FEM) are proposed in this volume. For a class of singularly perturbed problems on polygonal domains, robust exponential convergence of the hp-FEM based on these mesh design principles is established rigorously.

• FEM
• design
• elliptic regularity
• finite element method
• fluid mechanics
• high order method
• modeling
• non-smooth domains
• singular perturbation
• spectral method
• partial differential equations

Policies and ethics