Galerkin methods

Natural neighbour Galerkin methods

Natural neighbour co-ordinates (Sibson co-ordinates) is a well-known interpolation scheme for multivariate data tting and smoothing. The numerical implementation of natural neighbour co-ordinates in a Galerkin method is known as the natural element method (NEM). In the natural element method, natural neighbour co-ordinates are used to construct the trial and test functions. Recent studies on NE...

Adaptive Fourier-Galerkin methods

We study the performance of adaptive Fourier-Galerkin methods in a periodic box in R d with dimension d ≥ 1. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the hp-FEM. We examine two nonlinear approximation classes, one classical corresponding to alge...

Discontinuous Galerkin methods

This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience.We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another co...

Cell centered Galerkin methods

Plane Wave Discontinuous Galerkin Methods

Standard low order Lagrangian finite element discretization of boundary value problems for the Helmholtz equation −∆u−ωu = f are afflicted with the so-called pollution phenomenon: though for sufficiently small hω an accurate approximation of u is possible, the Galerkin procedure fails to provide it. Attempts to remedy this have focused on incorporating extra information in the form of plane wav...