A posteriori error estimates for the virtual element method

An a posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing

Complexity Analysis of a Fast Directional Matrix-Vector Multiplication

We consider a fast, data-sparse directional method to realize matrix-vector products related to point evaluations of the Helmholtz kernel. The method is based on a hierarchical partitioning of the point sets and the matrix. The considered directional multi-level approximation of the Helmholtz kernel can be applied even on high-frequency levels efficiently. We provide a detailed analysis of the almost linear asymptotic complexity of the presented method. Our numerical experiments are in good agreement with the provided theory.Comment: 20 pages, 2 figures, 1 tabl

A Geometrical Approach to the Boundary Element Method

We introduce a geometric formulation of the boundary element method (BEM), using concepts of the discrete electromagnetic theory. Geometric BEM is closely related to Galerkin-BEM and to the generalized collocation scheme. It is easy to implement, accurate, and computationally efficient. We validate our approach with 2-D examples and give an outlook to 3-D results