Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations

We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial meshes in two and three dimensions. We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error. The estimators are also shown to provide local lower bounds for the HDG energy (semi)norm of the error up to a constant and a higher-order data oscillation term. For the primal HDG methods and mixed HDG methods with an appropriate choice of stabilization parameter, the estimators are also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and a higher-order data oscillation term. Numerical examples are given illustrating the theoretical results

Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90641/1/AIAA-53965-537.pd

Algebraic and discretization error estimation by equilibrated fluxes for discontinuous Galerkin methods on nonmatching grids

International audienceWe derive a posteriori error estimates for the discontinuous Galerkin method applied to the Poisson equation. We allow for a variable polynomial degree and simplicial meshes with hanging nodes and propose an approach allowing for simple (nonconforming) flux reconstructions in such a setting. We take into account the algebraic error stemming from the inexact solution of the associated linear systems and propose local stopping criteria for iterative algebraic solvers. An algebraic error flux reconstruction is introduced in this respect. Guaranteed reliability and local efficiency are proven. We next propose an adaptive strategy combining both adaptive mesh refinement and adaptive stopping criteria. At last, we detail a form of the estimates avoiding any practical reconstruction of a flux and only working with the approximate solution, which simplifies greatly their evaluation. Numerical experiments illustrate a tight control of the overall error, good prediction of the distribution of both the discretization and algebraic error components, and efficiency of the adaptive strategy

Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors

We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.Comment: submitte

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling

International audienceWe consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size h , whereas the interfaces are partitioned by a mesh of much coarser size H , and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio H/h under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement strategy is proposed and numerically tested