Interior a posteriori error estimates for time discrete approximations of parabolic problems

a posteriori error estimates for time discrete approximations o

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L∞(0, T; L2(Ω)) and the higher order spaces, L∞(0, T;H1(Ω)) and H1(0, T; L2(Ω)), with optimal orders of convergence

On atomistic-to-continuum couplings without ghost forces in three dimensions

In this paper we construct energy based numerical methods free of ghost forces in three dimen- sional lattices arising in crystalline materials. The analysis hinges on establishing a connection of the coupled system to conforming finite elements. Key ingredients are: (i) a new representation of discrete derivatives related to long range interactions of atoms as volume integrals of gradients of piecewise linear functions over bond volumes, and (ii) the construction of an underlying globally continuous function representing the coupled modeling method

A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\Omega})) in parabolic problems

We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result with the residual based estimators from the aposteriori esti- mation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the L \infty (0, T ; L2({\Omega})) norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson (1991) by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estima- tors. For comparison with previous results we derive also an energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error which simplifies a previous one given in Lakkis and Makridakis (2006). We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.Comment: 30 pages, including 7 color plates in 4 figure

On the Babuška-Osborn approach to finite element analysis: L2 estimates for unstructured meshes

The standard approach to L2 bounds uses theH1 bound in combination to a duality argument, known as Nitsche’s trick, to recover the optimal a priori order of the method. Although this approach makes perfect sense for quasi-uniform meshes, it does not provide the expected information for unstructured meshes since the final estimate involves the maximum mesh size. Babuška and Osborn, [1], addressed this issue for a one dimensional problem by introducing a technique based on mesh-dependent norms. The key idea was to see the bilinear form posed on two different spaces; equipped with the mesh dependent analogs of L2 and H2 and to show that the finite element space is inf-sup stable with respect to these norms. Although this approach is readily extendable to multidimensional setting, the proof of the inf-sup stability with respect to mesh dependent norms is known only in very limited cases. We establish the validity of the inf-sup condition for standard conforming finite element spaces of any polynomial degree under certain restrictions on the mesh variation which however permit unstructured non quasiuniform meshes. As a consequence we derive L2 estimates for the finite element approximation via quasioptimal bounds and examine related stability properties of the elliptic projection

A posteriori L∞(L^2)-error bounds in finite element approximation of the wave equation

a

A posteriori error estimates for discontinuous Galerkin Methods for the Generalised Korteweg-de Vries Equation

We construct, analyze and numerically validate a posteriori error estimates for conservative discontinuous Galerkin (DG) schemes for the Generalized Korteweg-de Vries (GKdV) equation. We develop the concept of dispersive reconstruction, i.e., a piecewise polynomial function which satisfies the GKdV equation in the strong sense but with a computable forcing term enabling the use of a priori error estimation techniques to obtain computable upper bounds for the error. Both semidiscrete and fully discrete approximations are treated

Estimativos del error a posteriori para problemas de valores iniciales no lineales en el contexto de los espacios de Banach y los semigrupos

Un discretización en tiempo basada en el método de Euler regresivo para el problema parab´olico no lineal abstracto u = F(u), u(0) = u0, es considerada. En el presente trabajo se obtienen estimativos a posteriori para la citada discretización en tiempo en el marco de los espacios de Banach, los semigrupos y la regularidad maximal. Los estimativos obtenidos resultan ser de tipo condicional, es decir están sujetos a hipótesis que son verificables en la práctica como son las condiciones sobre la propia solución numérica