This paper studies the discontinuous Petrov–Galerkin (DPG) method, where the test space is normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter. The main finding is that as the parameter approaches zero, better results are obtained, under some circumstances, when the method is applied to the Helmholtz equation. Th...

A Discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi-discrete and a family of fully-discrete time approximate schemes are formulated. These schemes are symmetric. Hp-version error estimates are analyzed for these schemes. For the semi-discrete time scheme a priori L∞(H 1) error estimate is derived and similarly, l∞(H 1) and l2(H 1) for the...

Efficiently direct solvers based on the Jacobi–Galerkin method for the integrated forms of second-order elliptic equations in one and two space variables are presented. They are based on appropriate base functions for the Galerkin formulation which lead to discrete systems with specially structured matrices that can be efficiently inverted. The homogeneous Dirichlet boundary conditions are sati...

A discontinuous Galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions. This approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent to the discontinuous Galerkin method. Test problems include both ...

This paper provides equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green’s function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-di...

Matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the num...

There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green’s function is positive for triangular meshes allowing sufficiently good approximation of H1 functions. We study this question ...

We propose and analyse a new finite element method for convection diffusion problems based on the combination of a mixed method for the elliptic and a discontinuous Galerkin method for the hyperbolic part of the problem. The two methods are made compatible via hybridization and the combination of both is appropriate for the solution of intermediate convection-diffusion problems. By construction...

We propose and analyse a new finite element method for convection diffusion problems based on the combination of a mixed method for the elliptic and a discontinuous Galerkin method for the hyperbolic part of the problem. The two methods are made compatible via hybridization and the combination of both is appropriate for the solution of intermediate convection-diffusion problems. By construction...