Sparse p-version BEM for first kind boundary integral equations with random loading

We consider the weakly singular boundary integral equation V u = g ( ω ) on a deterministic smooth closed curve Γ ⊂ R 2 with random loading g ( ω ) . Given the kth order statistical moment of g, the aim is the efficient deterministic computation of the kth order statistical moment of u. We derive a deterministic formulation for the kth statistical moment. It is posed in the tensor product Sobolev space and involves the k-fold tensor product operator ⊗ i = 1 k V . The standard full tensor product Galerkin BEM requires O ( N k ) unknowns for the kth moment problem, where N is the number of unknowns needed to discretize Γ. Extending ideas of [V.N. Temlyakov, Approximation of functions with bounded mixed derivative, Proc. Steklov Inst. Math. (1989) vi+121. A translation of Trudy Mat. Inst. Steklov 178 (1986)], we develop the p-Sparse Grid Galerkin BEM to reduce the number of unknowns from O ( N k ) to O ( N ( log N ) k − 1 ) .