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Quasi-Monte Carlo finite element methods for elliptic PDEs with log-normal random coefficient
(2013)SAM Research ReportIn this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in $R^d$ (d=1,2,3), with diffusion coefficient a(x,ω) given as a lognormal random field, i.e., a(x,ω)=exp(Z(x,ω)) where x is the spatial variable and Z(x,⋅) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ over all possible realizations of ...Report -
Sparse tensor spherical harmonics approximation in radiative transfer
(2010)SAM Research ReportThe stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, inflow boundary conditions have to be prescribed. The sparse tensor product discretization has been successfully applied to finite element methods in radiative transfer with wavelet discretization of the angular domain (Widmer2009a). In this report we show that the ...Report -
Covariance structure of parabolic stochastic partial differential equations
(2012)SAM Research ReportIn this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space-time weak variational formulation of this tensorized equation is established.Report -
hp-FEM for second moments of elliptic PDEs with stochastic data Part 1: Analytic regularity
(2010)SAM Research ReportFor a linear second order elliptic partial differential operator $A: V → V'$, we consider the boundary value problems $Au=f$ with stationary Gaussian random data $f$ over the dual $V'$ of the separable Hilbert space $V$ in which the solution u is sought. The operator $A$ is assumed to be deterministic and bijective. The unique solution $u= A^-$$^1f $ is a Gaussian random field over $V$. It is characterized by its mean field $E_u$ and ...Report -
hp-FEM for second moments of elliptic PDEs with stochastic data Part 2: Exponential convergence
(2010)SAM Research ReportWe prove exponential rates of convergence of a class of $hp$ Galerkin Finite Element approximations of solutions to a model tensor non-hypoelliptic equation in the unit square □ = (0,1)$^2$ which exhibit singularities on ∂□ and on the diagonal ∆ = {($x,y$) ∈ □ : $x$ = $y$}, but are otherwise analytic in □. As we explained in the first part [6] of this work, such problems arise as deterministic second moment equations of linear, second ...Report -
Rapid solution of first kind boundary integral equations in R³
(2002)SAM Research ReportWeakly 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 ...Report -
hp Discontinuous Galerkin Time Stepping for Parabolic Problems
(2000)SAM Research ReportThe algorithmic pattern of the hp Discontinuous Galerkin Finite Element Method (DGFEM) for the time semidiscretization of abstract parabolic evolution equations is presented. In combination with a continuous $hp$ discretization in space we present a fully discrete hp-scheme for the numerical solution of parabolic problems. Numerical examples for the heat equation in a two dimensional domain confirm the exponential convergence rates which ...Report -
Wavelet-discretizations of parabolic integro-differential equations
(2001)SAM Research ReportWe 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 ...Report -
Mixed hp-finite element approximations on geometric edge and boundary layer meshes in three dimensions
(2001)SAM Research ReportIn this paper, we consider the Stokes problem in a three-dimensional polyhedral domain discretized with hp finite elements of type Qk for the velocity and Qk-2 for the pressure, defined on hexahedral meshes anisotropically and non quasi-uniformly refined towards faces, edges, and corners. The inf-sup constant of the discretized problem is independent of arbitrarily large aspect ratios and exhibits the same dependence on k as in in the ...Report -
Advanced boundary element algorithms
(1999)SAM Research ReportWe review recent algorithmic developments in the boundary element method (BEM) for large scale engineering calculations. Two classes of algorithms, the clustering and the wavelet-based schemes are compared. Both have $O(N(\log N)^a)$ complexity with some small $a \ge 0$ and allow in-core simulations with up to $N = O(10^6)$ DOF on the boundary on serial workstations. Clustering appears more robust for complex surfaces.Report