Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM

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

Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM

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

Anisotropic finite elements for the Stokes problem: a posteriori error estimator and adaptive mesh

AbstractWe propose an a posteriori error estimator for the Stokes problem using the Crouzeix–Raviart/P0 pair. Its efficiency and reliability on highly stretched meshes are investigated. The analysis is based on hierarchical space splitting whose main ingredients are the strengthened Cauchy–Schwarz inequality and the saturation assumption. We give a theoretical proof of a method to enrich the Crouzeix–Raviart element so that the strengthened Cauchy constant is always bounded away from unity independently of the aspect ratio. An anisotropic self-adaptive mesh refinement approach for which the saturation assumption is valid will be described. Our theory is confirmed by corroborative numerical tests which include an internal layer, a boundary layer, a re-entrant corner and a crack simulation. A comparison of the exact error and the a posteriori one with respect to the aspect ratio will be demonstrated