Element Methods For Hyperbolic Problems Research Articles

Some Observations on the Interaction Between Linear and Nonlinear Stabilization for Continuous Finite Element Methods Applied to Hyperbolic Conservation Laws

We discuss stability and accuracy of stabilized finite element methods for hyperbolic problems. In particular we focus on the interaction between linear and nonlinear stabilizations. First we show that the combination of linear and nonlinear stabilization can be designed to be invariant preserving. Then we show that such a combined method allows for the classical error estimates for smooth solutions of space semidiscretized formulations of the linear transport equation. Based on these ideas we propose a Runge–Kutta finite element method for the compressible Euler equations using entropy viscosity to ensure stability at shocks and gradient jump penalty to prevent propagation of high frequency content into the smooth part of the solution. In a numerical example we show that the method predicts the shock structure accurately, without high frequency pollution of the smooth parts.

Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems

We consider a discontinuous Galerkin finite element method for the advection–reaction equation in two space–dimensions. For polynomial approximation spaces of degree greater than or equal to two on triangles we propose a method where stability is obtained by a penalization of only the upper portion of the polynomial spectrum of the jump of the solution over element edges. We prove stability in the standard h-weighted graphnorm and obtain optimal order error estimates with respect to mesh-size.

A mortar element method for hyperbolic problems

A non-conforming finite element method based on non-overlapping domain decomposition is extended to linear hyperbolic problems. The method is based on streamline-diffusion/discontinuous Galerkin methods and the mortar element method. A weak flux continuity condition at the inflow interface is enforced by means of Lagrange multipliers. This weak flux continuity condition replaces the usual mortar condition for elliptic problems, and allows non-matching grids at the subdomain interfaces. To cite this article: Y. Bourgault, A. El Boukili, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Hp‐Discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity

AbstractIn this paper we develop the a posteriori error analysis of the hp‐version of the discontinuous Galerkin finite element method for linear and non‐linear hyperbolic problems. By employing a duality argument, sharp a posteriori error bounds are derived for certain output functionals of practical interest. These bounds exhibit an exponential rate of convergence under hp‐refinement if either the primal or the dual solution is an analytic function over the computational domain. Based on our a posteriori error bounds, we design and implement the corresponding hp‐adaptive finite element algorithm to ensure the reliable and efficient control on the error in the prescribed functional to within a user‐defined tolerance. Copyright © 2002 John Wiley & Sons, Ltd.

On the stability of a family of finite element methods for hyperbolic problems

We consider a family of tensor product finite element methods for hyperbolic equations in RN, N ≥ 2, which are explicit and generate a continuous approximate solution. The base case N = 2 (an extension of the box scheme to higher order) is due to Winther, who proved stability and optimal order convergence. By means of a simple counterexample, we show that, for linear approximation with N ≥ 3, the corresponding methods are unstable.

An Analysis of Some High Accuracy Finite Element Methods for Hyperbolic Problems

Some high accuracy finite element methods for hyperbolic problems are studied in this paper. It is proven that over a finite element mesh, the convergence order of linear finite element solutions for both linear and nonlinear equations can be higher than one even if the exact solutions are discontinuous. The theoretical tools for the convergence analysis are some superclose error estimates that are also developed in this paper for nonsmooth solutions.

SOME ARGUMENTS FOR RECOVERING THE FINITE ELEMENT

The dual argument is well known for recoving the optimal L2-error of the finite element method in elliptic context. This argument, however, will lose efficacy in hyperbolic case. An expansion argument and an approximation argument are presented in this paper to recover the optimal L2-error of finite element methods for hyperbolic problems. In particular, a second order error estimate in L2-norm for the standard linear finite element method of hyperbolic problems is obtained if the exact solution is smooth and the finite element mesh is almost uniform, and some superconvergence estimates are also established for less smooth solution.