Anisotropic Finite Elements: Local Estimates and Applications (Advances in Numerical Mathematics) - Softcover

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Anisotropic finite element meshes have different mesh sizes in different directions. Such meshes have a great potential for the approximation of functions with anisotropic behaviour, as for example near edges or in boundary layers. The aim of this monograph is to present a mathematical theory of the approximation properties of finite element spaces over anisotropic meshes. Local error estimates are derived for the Lagrange interpolation and for modified Scott-Zhang interpolation operators. Families of anisotropic finite element meshes are constructed for the numerical solution of model problems with boundary layers or edge and corner singularities, and the global discretization error is estimated. Numerical testes show that the asymptotic results are valid for a moderate number of unknowns already. The strengths of this investigation are the consideration of two- and three-dimensional problems in general polygonal/polyhedral domains, and the treatment of lower and higher of finite elements.