Max Jensen

A unifying theory of a posteriori error control for discontinuous Galerkin FEM

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617–637, 2005) and Carstensen and Hu (J Numer Math 107(3):473–502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes.

hp -Discontinuous Galerkin Finite Element Methods with Least-Squares Stabilization

We consider a family of hp-version discontinuous Galerkin finite element methods with least-squares stabilization for symmetric systems of first-order partial differential equations. The family includes the classical discontinuous Galerkin finite element method, with and without streamline-diffusion stabilization, as well as the discontinuous version of the Galerkin least-squares finite element method. An hp-optimal error bound is derived in the associated DG-norm. If the solution of the problem is elementwise analytic, an exponential rate of convergence under p-refinement is proved. We perform numerical experiments both to illustrate the theoretical results and to compare the various methods within the family.

On the convergence of finite element methods for Hamilton-Jacobi-Bellman equations

We study the convergence of monotone

Discontinuous Galerkin Finite Element Convergence for Incompressible Miscible Displacement Problems of Low Regularity

P1

Frontiers in numerical analysis - Durham 2010

finite element methods on unstructured meshes for fully nonlinear Hamilton--Jacobi--Bellman equations arising from stochastic optimal control problems with possibly degenerate, isotropic diffusions. Using elliptic projection operators we treat discretizations which violate the consistency conditions of the framework by Barles and Souganidis. We obtain strong uniform convergence of the numerical solutions and, under nondegeneracy assumptions, strong

On the Stability of Continuous---Discontinuous Galerkin Methods for Advection---Diffusion---Reaction Problems

L^2

DISCONTINUOUS GALERKIN METHODS FOR MASS TRANSFER THROUGH SEMIPERMEABLE MEMBRANES

convergence of the gradients.

Convergent Semi-Lagrangian Methods for the Monge--Ampère Equation on Unstructured Grids

In this article we analyze the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous Galerkin method under minimal regularity assumptions. The main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem. In order to deal with the nonconformity of the method and to avoid overpenalization of jumps across interelement boundaries, the careful construction of a reflexive subspace of the space of bounded variation, which compactly embeds into

Stable Crank-Nicolson Discretisation for Incompressible Miscible Displacement Problems of Low Regularity

L^2(\Omega)

Finite Element Methods with Artificial Diffusion for Hamilton-Jacobi-Bellman Equations

, and of a lifting operator, which is compatible with the nonlinear diffusion coefficient, are required. An equivalent skew-symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows us to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method. Numerical experiments underline the robustness of the proposed algorithm.