Homogeneous, linear elliptic boundary value problems with constant coefficients can be transformed into boundary integral equations by using the integral equation method. In this chapter we will introduce the relevant boundary integral operators and we will derive the most important mapping properties and representations. We will also present the boundary integral equations for the boundary value problems from the previous chapter. Finally, we will prove the appropriate results on existence and uniqueness for these boundary integral equations. Boundary Integral OperatorsWe consider the differential operator L from (2.98)Our goal is to solve the homogeneous differential equationfor this operator with appropriate boundary conditions. Solutions of these differential equations can be formulated with the help of potentials that are closely linked to the fundamental solution of the operator L, which in turn can be formulated explicitly. In general, we assume that the coefficients of L satisfy A 2 R We set # WD c C kbk 2

Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDE's in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space V = H 1 0 (D) of weak solutions of the elliptic problem with a controlled number of terms N . The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coefficients from a family of continuous, piecewise linear finite element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number N dof of degrees of freedom is the minimum of the convergence rates afforded by the best N -term sequence approximations in the parameter space and the rate of finite element approximations in D for a single instance of the parametric problem.

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Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D ⊂ R d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2 (D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y = y(ω) = (y i (ω)). This yields an equivalent parametric deterministic PDE whose solution u(x, y) is a function of both the space variable x ∈ D and the in general countably many parameters y. Communicated by Wolfgang Dahmen. 616 Found Comput Math (2010) 10: 615-646We establish new regularity theorems describing the smoothness properties of the solution u as a map from y ∈ U = (−1, 1) ∞ to V = H 1 0 (D). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called "generalized polynomial chaos" (gpc) expansion of u.Convergence estimates of approximations of u by best N -term truncated V valued polynomials in the variable y ∈ U are established. These estimates are of the form N −r , where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N "samples" (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients.A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family {V l } ∞ l=0 ⊂ V of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x, y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y ∈ U = (−1, 1) ∞ to a smoothness space W ⊂ V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with H 2 (D) ∩ H 1 0 (D) in the case where D is a smooth or convex domain.Our analysis shows that in realistic settings a convergence rate N −s dof in terms of the total number of degrees of freedom N dof can be obtained. Here the rate s is determined by both the best N -term approximation rate r and the approximation order of the space discretization in D.

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