In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier–Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint sec...

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the L∞ norm. The discretization method uses of continuous Lagrange finite elements in space and discontinuous Galerkin methods in time of an arbitrary order. The method of proof differs from the established ...

In this work we extend to the Stokes problem the Discontinuous Galerkin Reduced Basis Element (DGRBE) method introduced in [1]. By this method we aim at reducing the computational cost for the approximation of a parametrized Stokes problem on a domain partitioned into subdomains. During an offline stage, expensive but performed only once, a low-dimensional approximation space is built on each s...

We develop a class of stochastic numerical schemes for Hamilton-Jacobi equations with random inputs in initial data and/or the Hamiltonians. Since the gradient of the HamiltonJacobi equations gives a symmetric hyperbolic system, we utilize the generalized polynomial chaos (gPC) expansion with stochastic Galerkin procedure in random space and the JinXin relaxation approximation in physical space...

In this paper, a rigorous convergence and error analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem is presented. The formulation of the eigenvalue problem in terms of a boundary integral equation yields a nonlinear boundary integral operator eigenvalue problem. This nonlinear eigenvalue problem and its Galerkin approximation are analyzed in the framewo...

Starting from the generalized Lax-Milgram theorem and from the fact that the approximation error is minimized when the continuity and inf– sup constants are unity, we develop a theory that provably delivers well-posed approximation methods with unity continuity and inf–sup constants for numerical solution of linear partial differential equations. We demonstrate our single-framework theory on sc...

The local discontinuous Galerkin method for the numerical approximation of the time-harmonic Maxwell equations in a low-frequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The presented method involves discontinuous Galerkin discretizations of the curl-curl and grad-div operato...