In this paper we study the two-dimensional version of the RungeKutta Local Projection Discontinuous Galerkin (RKDG) methods, already defined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-order accurate. Preliminary numerical results showing ...

Modeling the interaction between surface and subsurface flow is a challenging environmental problem. One such example is the simulation of transport of contaminants through rivers into the aquifers. Mathematically, this complex problem can be modeled by the coupled system of Stokes and Darcy equations. The aim of this paper is to formulate and analyze a discontinuous finite element method for t...

In this article we develop and analyze two-level and multi-level methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coefficients (exhibiting large jumps across interfaces in the domain). These methods are based on a decomposition of the DG finite element space that inherently hinges on the diffusion coefficien...

This is a further development of [10] regarding multilevel preconditioning for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. We assume that the mesh on the finest level is a results of a geometrically refined fixed coarse mesh. The preconditioner is a multilevel method that uses a sequence of finite element spaces of either co...

We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by C(t − s)α−1, where 0 < α < 1. For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most q − 1, for q = 1 or 2. For the space discretization we use continuous piecewise-linear finite elements. The d...

We consider a family of mixed finite element discretizations of the Darcy flow equations using totally discontinuous elements (both for the pressure and the flux variable). Instead of using a jump stabilization as it is usually done for DG methods (see e.g. [3], [13] and the references therein) we use the stabilization introduced in [18], [17]. We show that such stabilization works for disconti...

Spacetime discontinuous Galerkin finite element methods [1–3] rely on ‘target fluxes’ on elementboundaries that are computed via local one-dimensional Riemann solutions in the direction normal toelement face. In this work, we demonstrate a generalized solution procedure for linearized hyperbolicsystems based on diagonalisation of the governing system of partial differential equa...

A discontinuous Galerkin method, with hp-adaptivity based on the approximate solution of appropriate dual problems, is employed for highly-accurate eigenvalue computations on a collection of benchmark examples. After demonstrating the effectivity of our computed error estimates on a few well-studied examples, we present results for several examples in which the coefficients of the partial-diffe...

Abstract. This paper is concerned with the analysis of the discontinuous Galerkin finite element method applied to nonstationary convection-diffusion problems with nonlinear convection and nonlinear diffusion. We generalize results from [2], where linear diffusion is assumed. Optimal error estimates are obtained for the L(H) norm and interelement jump terms, however due to the nonlinearity of t...

Glossary Numerical methods Processes in nature are often described by partial differential equations. Finding solutions to those equations is at the heart of many studies aiming at the explanation of observed data. Simulations of realistic physical processes requires generally the use of numerical methods – a special branch of applied mathematics – that approximate the partial differential equa...