Averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing. In the second paper of our analysis of their reliability, we consider conforming h-FEM of higher (i.e., not of lowest) order in two or three space dimensions. In this paper, reliablility is shown for conforming higher order finite elemen...

The celebrated Spectral Element Method (SEM) has appeared to be of great interest to extend the capabilities of spectral methods to complex geometries. However, using quadrangular (hexahedral in 3D) elements may be a severe restriction for the most complex geometries, because requiring a structured mesh. Thus, Finite element meshes are usually based on triangles (tetrahedra in 3D). Efforts have...

We present 2-D, 3-D, and spherical mesh generators for the Finite Element Method (FEM) using triangular and tetrahedral elements. The mesh nodes are treated as if they were linked by virtual springs that obey Hooke’s law. Given the desired length for the springs, the FEM is used to solve for the optimal nodal positions for the static equilibrium of this spring system. A ’guide-mesh’ approach al...

We propose employing the extension of the Lehmann-MaehlyGoerisch method developed by Zimmermann and Mertins, as a highly effective tool for the pollution-free finite element computation of the eigenfrequencies of the resonant cavity problem on a bounded region. This method gives complementary bounds for the eigenfrequencies which are adjacent to a given parameter t ∈ R. We present a concrete nu...

A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is developed. It combines the best features of structured and unstructured grid methods to attain computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finit...

A new agglomeration multigrid method is proposed in this paper for general unstructured grids. By a proper local agglomeration of finite elements, a nested sequence of finite dimensional subspaces are obtained by taking appropriate linear combinations of the basis functions from previous level of space. Our algorithm seems to be able to solve, for example, the Poisson equation discretized on an...