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The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and the right-hand side function depend on infinitely many (uncertain) parameters. We introduce a two-level a posteriori estimator to control the energy error in multilevel stochastic Galerkin approximations for this class of PDE problems. We prove that the two-level estimator always provides a lower bound for the unknown approximation error, while the upper bound is equivalent to a saturation assumption. We propose and empirically compare three adaptive algorithms, where the structure of the estimator is exploited to perform spatial refinement as well as parametric enrichment. The paper also discusses implementation aspects of computing multilevel stochastic Galerkin approximations.
Partial differential equations (PDEs) with inputs that depend on infinitely many parameters pose serious theoretical and computational challenges. Sophisticated numerical algorithms that automatically determine which parameters need to be activated i
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element ap
The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field PDE model in which the Youngs modulus is an affine function of a co
Hybrid quantum/molecular mechanics models (QM/MM methods) are widely used in material and molecular simulations when MM models do not provide sufficient accuracy but pure QM models are computationally prohibitive. Adaptive QM/MM coupling methods feat
Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coeffic