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Sparse Finite Elements for Elliptic Problems with Stochastic Data
(2002)SAM Research ReportWe formulate elliptic boundary value problems with stochastic loading in a domain D. We show well-posedness of the problem in stochastic Sobolev spaces and we derive then a deterministic elliptic PDE in DxD for the spatial correlation of the solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobolev spaces with mixed highest order derivatives. Discretization with sparse tensor products of any ...Report -
Mixed hp-DGFEM for incompressible flows II: Geometric edge meshes
(2002)SAM Research ReportWe consider the Stokes problem in three-dimensional polyhedral domains discretized on hexahedral meshes with hp-discontinuous Galerkin finite elements of type IQk for the velocity and IQk-1 for the pressure. We prove that these elements are inf-sup stable on geometric edge meshes that are refined anisotropically and non quasi-uniformly towards edges and corners. The discrete inf-sup constant is shown to be independent of the aspect ratio ...Report -
Mixed hp-DGFEM for incompressible flows
(2002)SAM Research ReportWe 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.Report -
Mixed hp-DGFEM for incompressible flows III: Pressure stabilization
(2002)SAM Research ReportWe consider stabilized mixed hp-discontinuous Galerkin methods for the discretization of the Stokes problem in three-dimensional polyhedral domains. The methods are stabilized with a term penalizing the pressure jumps. For this approach it is shown that IQk-IQk and IQk-IQk-1 elements satisfy a generalized inf-sup condition on geometric edge and boundary layer meshes that are refined anisotropically and non quasi-uniformly towards faces, ...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 -
Analytic regularity of Stokes flow in polygonal domains
(2002)SAM Research ReportWe investigate the analytic regularity of the Stokes problem in a polygonal domain $\Omega \subset R^2$ with straight sides and piecewise analytic data. We establish a shift theorem in weighted Sobolev spaces of arbitrary order with explicit control of the order-dependence of the constants. The shift-theorem in countably normed weighted Sobolev spaces implies in particular interior analyticity and Gevrey-type analytic regularity near the corners.Report -
Multiresolution weighted norm equivalences and applications
(2002)SAM Research ReportWe establish multiresolution norm equivalences in weighted spaces $L^2_w$ ((0,1)) with possibly singular weight functions $w(x) \geq 0$ in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function $w(x)$ within each scale. Since norm equivalences for Sobolev norms are by now ...Report -
Fast deterministic pricing of options on Lévy driven assets
(2002)SAM Research ReportArbitrage-free prices $u$ of European contracts on risky assets whose logreturns are modelled by Lévy processes satisfy a parabolic parabolic partial integrodifferential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$. This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the $\theta$-scheme in time and a wavelet Galerkin method with $N$ degrees of freedom ...Report