Integral potential method for a transmission problem with Lipschitz interface in R^3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R3, one of them is a bounded Lipschitz domain with connected boundary, and the other one is the exterior Lipschitz domain R3 n. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces

Superconvergent Graded Meshes for an Elliptic Dirichlet Control Problem

Chemnitz Finite Element Symposium (30th. 2017. St. Wolfgang/Strobl, Austria

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator Delta :Hk+1(M) boolean AND H-0(1)((M) -> Hk-1 (M), k is an element of N-0, invertible?" We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103-120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let partial derivative M-D subset of partial derivative M be an open and closed subset of the boundary of M. We say that (M, partial derivative M-D) has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance dist (x, partial derivative M-D) from a point x subset of M to partial derivative M-D subset of partial derivative M is bounded uniformly in x (and hence, in particular, partial derivative M-D intersects all connected components of M). For manifolds (M, partial derivative M-D) with finite width, we prove a Poincare inequality for functions vanishing on partial derivative M-D, thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincare inequality then leads, as in the classical case to results on the spectrum of Delta with domain given by mixed boundary conditions, in particular, Delta is invertible for manifolds (M, partial derivative M-D) with finite width. The bounded geometry assumption then allows us to prove the well-posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces H-s(M), s >= 0

Maximal regularity of parabolic transmission problems

Linear reaction–diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal L_\mathrm{p} regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain transversally

Newtonian and single layer potentials for the Stokes system with L^∞ coefficients and the exterior Dirichlet problem

EPSRC, U

Multilevel Quasi-Monte Carlo Uncertainty Quantification for Advection-Diffusion-Reaction

We survey the numerical analysis of a class of deterministic, higher-order QMC integration methods in forward and inverse uncertainty quantification algorithms for advection-diffusion-reaction (ADR) equations in polygonal domains D⊂R2 with distributed uncertain inputs. We admit spatially heterogeneous material properties. For the parametrization of the uncertainty, we assume at hand systems of functions which are locally supported in D. Distributed uncertain inputs are written in countably parametric, deterministic form with locally supported representation systems. Parametric regularity and sparsity of solution families and of response functions in scales of weighted Kontrat’ev spaces in D are quantified using analytic continuation.ISSN:2194-1009ISSN:2194-101