Endre Süli

Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality

We give an overview of recent developments concerning the use of adjoint methods in two areas: the a posteriori error analysis of finite element methods for the numerical solution of partial differential equations where the quantity of interest is a functional of the solution, and superconvergent extraction of integral functionals by postprocessing.

Discontinuous hp -Finite Element Methods for Advection-Diffusion-Reaction Problems

We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, advection-reaction equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived; here, we consider only advection-reaction problems which satisfy a certain (standard) positivity condition. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

\frac{1}{2}

Convergence of finite volume schemes for Poisson's equation on nonuniform meshes

a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For elementwise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.

A convergence analysis of Yee's scheme on nonuniform grids

SummaryThe Lagrange-Galerkin method is a numerical technique for solving convection — dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method. We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.

DISCONTINUOUS GALERKIN METHODS FOR FIRST-ORDER HYPERBOLIC PROBLEMS

An error analysis of a family of cell-center finite volume schemes for Poisson’s equation on Cartesian product nonuniform meshes in two dimensions is presented. Optimal-order error estimates are derived in the discrete

hp -Adaptive Discontinuous Galerkin Finite Element Methods for First-Order Hyperbolic Problems

H^1

Stabilized hp -Finite Element Methods for First-Order Hyperbolic Problems

norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh. On quasi-uniform meshes analogous estimates are obtained in the maximum norm.

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

The Yee scheme is the principal finite difference method used in computing time domain solutions of Maxwell’s equations. On a uniform grid the method is easily seen to be second-order convergent in space. This paper shows that the Yee scheme is also second-order convergent on a nonuniform mesh despite the fact that the local truncation error is (nodally) only of first order.

A posteriori error analysis for stabilised finite element approximations of transport problems

The main aim of this paper is to highlight that, when dealing with DG methods for linear hyperbolic equations or advection-dominated equations, it is much more convenient to write the upwind numerical flux as the sum of the usual (symmetric) average and a jump penalty. The equivalence of the two ways of writing is certainly well known (see e.g. Ref. 4); yet, it is very widespread not to consider upwinding, for DG methods, as a stabilization procedure, and too often in the literature the upwind form is preferred in proofs. Here, we wish to underline the fact that the combined use of the formalism of Ref. 3 and the jump formulation of upwind terms has several advantages. One of them is, in general, to provide a simpler and more elegant way of proving stability. The second advantage is that the calibration of the penalty parameter to be used in the jump term is left to the user (who can think of taking advantage of this added freedom), and the third is that, if a diffusive term is present, the two jump stabilizations (for the generalized upwinding and for the DG treatment of the diffusive term) are often of identical or very similar form, and this can also be turned to the users advantage.