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Stochastic Finite Element Methods with the Euclidean Degree for Partial Differential Equations with Random Inputs

In this paper, we construct a new implementation of stochastic finite element methods for partial differential equations with random inputs. The basis functions of generalized polynomial chaos are determined not by the usual notion of degree of a multivariate polynomial, but by the Euclidean degree. Then the corresponding linear combination of basis from the stochastic finite element methods is obtained, where the coefficient matrix is sparse and symmetric. In numerical experiments considering stochastic diffusion and Helmholtz equations, our approach with Euclidean degree of gPC basis achieves a better convergence rate than ones with total degree.