hp-Finite Element Methods for Singular Perturbations [Paperback]

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.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.1.Introduction.- Part I: Finite Element Approximation.- 2. hp-FEM for Reaction Diffusion Problems: Principal Results.- 3. hp Approximation.- Part II: Regularity in Countably Normed Spaces.- 4. The Countably Normed Spaces blb,e.- 5. Regularity Theory in Countably Normed Spaces.- Part III: Regularity in Terms of Asymptotic Expansions.- 6. Exponentially Weighted Countably Normed Spaces.- Appendix.- References.- Index.Springer Book Archives