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Beschreibung:
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 (FEMs) for second order elliptic equations. Here, we revisit equilibrated fluxes for higher-order FEMs with nonconstant coefficients and illustrate the favorable 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 discretization, a novel a posteriori error estimator for the stochastic error in the energy norm is devised. It is based on the stochastic residual and its decomposition into approximation residuals and a truncation error of the stochastic discretization due to the reduction to a finite set of (random) parameters. 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 discretization in anisotropic Legendre polynomial chaos. Several stochastic benchmark problems illustrate the efficiency of the adaptive process.