The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems
Journal of Computational Physics
Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs
Foundations of Computational Mathematics
Numerical Methods for Stochastic Computations: A Spectral Method Approach
Numerical Methods for Stochastic Computations: A Spectral Method Approach
Sparse Tensor Discretization of Elliptic sPDEs
SIAM Journal on Scientific Computing
Editorial: High-order finite element approximation for partial differential equations
Computers & Mathematics with Applications
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Solutions of random elliptic boundary value problems admit efficient approximations by polynomials on the parameter domain. Each coefficient in such an expansion is a spatially dependent function, and can be approximated within a hierarchy of finite element spaces. If the finite elements are of sufficiently high order, using just a single spatial mesh is predicted to achieve the same convergence rate with respect to the total number of degrees of freedom as sparse tensor product constructions and other multilevel stochastic Galerkin approximations. Numerical computations for an elliptic two-point boundary value problem confirm this and indicate no loss of accuracy for a single-level method compared to using a sparse tensor product with the same total number of degrees of freedom.