Search

NQC

Basic convergence rates are established for an adaptive algorithm based on the dual weighted residual error representation, [GRAPHICS] applied to isoparametric d-linear quadrilateral finite element approximation of functionals of multi scale solutions to second order elliptic partial differential equations in bounded domains of R-d. In contrast to the usual aim to derive an a posteriori error estimate, this work derives, as the mesh size tends to zero, a uniformly convergent error expansion for the error density, with computable leading order term. It is shown that the optimal adaptive isotropic mesh uses a number of elements proportional to the d/2 power of the Ld/d+2 quasi-norm of the error density; the same error for approximation with a uniform mesh requires a number of elements proportional to the d/2 power of the larger L-1 norm of the same error density. A point is that this measure recognizes different convergence rates for multi scale problems, although the convergence order may be the same. The main result is a proof that the adaptive algorithm based on successive subdivisions of elements reduces the maximal error indicator with a factor or stops with the error asymptotically bounded by the tolerance using the optimal number of elements, up to a problem independent factor. An important step is to prove uniform convergence of the expansion for the error density, which is based on localized averages of second order difference quotients of the primal and dual finite element solutions. The averages are used since the difference quotients themselves do not converge pointwise for adapted meshes. The proof uses weak convergence techniques with a symmetrizer for the second order difference quotients and a splitting of the error into a dominating contribution, from elements with no hanging nodes or edges on the initial mesh, and a remaining asymptotically negligible part. Numerical experiments for an elasticity problem with a crack and different variants of the ave, QC 20100824

Convergence rates of adaptive algorithms for weak approximations of Ito stochastic differential equations are proved for the Monte Carlo Euler method. Two algorithms based either oil optimal stochastic time steps or optimal deterministic time steps are studied. The analysis of their computational complexity combines the error expansions with a posteriori leading order term introduced in Szepessy et al. [Szepessy, A.. R. Tempone, and G. Zouraris. 2001. Comm. Pare Appl. Math. 54:1169-1214] and ail extension of the convergence results for adaptive algorithms approximating deterministic ordinary differential equations, derived in Moon et al. [Moon, K.-S., A. Szepessy, R. Tempone, and G. Zouraris. 2003. Numer. Malh. 93:99-129]. The main step in the extension is the proof of the almost sure convergence of the error density. Both adaptive alogrithms are proven to stop with asymptotically optimal number of steps up to a problem independent factor defined in the algorithm. Numerical examples illustrate the behavior of the adaptive algorithms, motivating when stochastic and deterministic adaptive time steps are more efficient than constant time steps and when adaptive stochastic steps are more efficient than adaptive deterministic steps., QC 20100525

The theory of a posteriori error estimates suitable for adaptive refinement is well established. This work focuses on the fundamental, but less studied, issue of convergence rates of adaptive algorithms. In particular, this work describes a simple and general adaptive algorithm applied to ordinary, stochastic and partial differential equations with proven convergence rates. The presentation has three parts: The error approximations used to build error indicators for the adaptive algorithm are based on error expansions with computable leading order terms. It is explained how to measure optimal convergence rates for approximation of functionals of the solution, and why convergence of the error density is always useful and subtle in the case of stochastic and partial differential equations. The adaptive algorithm, performing successive mesh refinements, either reduces the maximal error indicator by a factor or stops with the error asymptotically bounded by the prescribed accuracy requirement. Furthermore, the algorithm stops using the optimal number of degrees of freedom, up to a problem independent factor., QC 20100824