Robust Adaptive $hp$ Discontinuous Galerkin Finite Element Methods for the Helmholtz Equation

Norm Preconditioners for Discontinuous Galerkin hp-Finite Element Methods

Abstract. We consider a norm-preconditioning approach for the solution of discontinuous Galerkin finite element discretizations of second order PDE with non-negative characteristic form. In particular, we perform an analysis for the general case of discontinuous hp-finite element discretizations. Our solution method is a norm-preconditioned threeterm GMRES routine. We find that for symmetric po...

hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number

This paper develops some interior penalty hp-discontinuous Galerkin (hp-DG) methods for the Helmholtz equation in two and three dimensions. The proposed hp-DG methods are defined using a sesquilinear form which is not only mesh-dependent but also degree-dependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the elemen...

Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows

Adaptive Galerkin Finite Element Methods for the Wave Equation

This paper gives an overview of adaptive discretization methods for linear second-order hyperbolic problems such as the acoustic or the elastic wave equation. The emphasis is on Galerkin-type methods for spatial as well as temporal discretization, which also include variants of the Crank-Nicolson and the Newmark finite difference schemes. The adaptive choice of space and time meshes follows the...

Finite Element and Discontinuous Galerkin Method for Stochastic Helmholtz Equation in Two- and Three-dimensions

Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using these so-called stochastic partial differential equations (SPDEs) is that SPDEs are able to more fully capture the behavior of interesting phenomena; it also means that the corresponding numerical analysis of the model will require...