We consider several mixed discontinuous Galerkin approximations of the Stokes problem and propose an abstract framework for their analysis. Using this framework we derive a priori error estimates for hp-approximations on tensor product meshes. We also prove a new stability estimate for the discrete divergence bilinear form.

Weakly singular boundary integral equations $(BIEs)$ of the first kind on polyhedral surfaces $\Gamma$ in $R^3$ are discretized by Galerkin BEM on shape-regular, but otherwise unstructured meshes of meshwidth $h$. Strong ellipticity of the integral operator is shown to give nonsingular stiffness matrices and, for piecewise constant approximations, up to $O(h^3)$ convergence of the farfield. The condition number of the stiffness matrix ...

We consider parabolic problems u + Au = f in (0,T)x Ω, T < ∞, where Ω ⊂ Rd is a bounded domain and A is a strongly elliptic, classical pseudo-differential operator of order ρ ∈ [0,2] in H ρ/2 (Ω). We use a θ-scheme for time discretization and a Galerkin method with N degrees of freedom for space discretization. The full Galerkin matrix for A can be replaced with a sparse matrix using a wavelet basis, and the linear systems for each time ...

We introduce the concept of generalized Finite Element Method (gFEM) for the numerical treatment of homogenization problems. These problems are characterized by highly oscillatory periodic (or patchwise periodic) pattern in the coefficients of the differential equation and their solutions exhibit a multiple scale behavior: a macroscopic behavior superposed with local characteristics at micro length scales. The gFEM is based on two-scale ...

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D_Rd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(!)=(yi(!)). This yields an equivalent parametric deterministic PDE whose ...