In a previous paper a preconditioning strategy based on overlapping domain decomposition was applied to the Galerkin approximation of elliptic partial differential equations on the sphere. In this paper the methods are extended to more general pseudodifferential equations on the sphere, using as before spherical radial basis functions for the approximation space, and again preconditioning the i...

Complementary problems play a central role in equilibrium finding, physical simulation, and optimization. As a consequence, we are interested in understanding how to solve these problems quickly, and this often involves approximation. In this paper we present a method for approximately solving strictly monotone linear complementarity problems with a Galerkin approximation. We also give bounds f...

We present and analyze reliable numerical techniques for simulating complex flow and transport phenomena related to natural gas transportation in pipelines. Such kind of problems are of high interest in the field of petroleum and environmental engineering. Modeling and understanding natural gas flow and transformation processes during transportation is important for the sake of physical realism...

It is shown that for a broad class of equations that numerical solutions computed using the discontinuous Galerkin or the continuous Galerkin time stepping schemes of arbitrary order will inherit the compactness properties of the underlying equation. Convergence of numerical schemes for a phase field approximation of the flow of two fluids with surface tension is presented to illustrate these r...

Abstract. We present a discontinuous Galerkin scheme for the numerical approximation of the onedimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semi-Lagrangian method to emphasize the good behavior of this scheme when applie...

A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary valué problems. The class of Galerkin spaces considered includes almost all (quasiuniform) piecewise-polynomial spaces that are used in practice. The estimates are optimal in the sense that no better rate of approximation is possible in general in the spaces employed.

We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We start by considering adaptive Fourier-Galerkin methods and Legendre spectralelement methods, which offer unlimited approximation power only restricted by solution and data regularity. We next turn our attention to the $hp$-version of the finite element method, for which we...

We give a space-time Galerkin nite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial diierential equation with a Volterra (memory) term. The discretization consists of a continuous piecewise linear approximation in space with a discontinuous piecewise constant or linear approximation in time. We derive an a priori maximum-e...