Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Doi
Abstract
Equilibration error estimators have been shown to commonly lead to very
accurate guaranteed error bounds in the a posteriori error control of finite
element methods for second order elliptic equations. Here, we extend previous
results by the design of equilibrated fluxes for higher-order finite element
methods with nonconstant coefficients and illustrate the favourable
performance of different variants of the error estimator within two
deterministic benchmark settings. After the introduction of the respective
parametric problem with stochastic coefficients and the stochastic Galerkin
FEM discretisation, a novel a posteriori error estimator for the stochastic
error in the energy norm is devised. The error estimation is based on the
stochastic residual and its decomposition into approximation residuals and a
truncation error of the stochastic discretisation. Importantly, by using the
derived deterministic equilibration techniques for the approximation
residuals, the computable error bound is guaranteed for the considered class
of problems. An adaptive algorithm allows the simultaneous refinement of the
deterministic mesh and the stochastic discretisation in anisotropic Legendre
polynomial chaos. Several stochastic benchmark problems illustrate the
efficiency of the adaptive process