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High order Galerkin approximations for parametric second order elliptic partial differential equations

by V. Nistor and Ch. Schwab

(Report number 2012-21)

Abstract
Let \(D \subset IR^d\), \(d = 2, 3\), be a bounded domain with piecewise smooth boundary \(\partial D\) and let \(U\) be an open subset of a Banach space \(Y\). We consider a parametric family \(P_y\) of uniformly strongly elliptic, parametric second order partial differential operators \(P_y\) on \(D\) in divergence form, where the parameter \(y\) ranges in the parameter domain \(U\) so that, for a given set of data \(f_y\), the solution \(u\) and the coefficients of the parametric boundary value problem \(P_y u = f_y\) are functions of \((x, y) \in D \times U\). Under suitable regularity assumptions on these coefficients and on the source term \(f\), we establish a regularity result for the solution \(u : D \times U \rightarrow IR\) of the parametric, elliptic boundary value problem \(P_y u(x, y) = f_y (x) = f (x, y)\), \(x \in D\), \(y \in U\), with mixed Dirichlet-Neumann boundary conditions. Let \(\partial D = \partial_d D \cup \partial_n D\) denote decomposition of the boundary into a part on which we assign Dirichlet boundary conditions and the part on which we assign Neumann boundary conditions. We assume that \(\partial_d D\) is a finite union of closed polygonal subsets of the boundary such that no adjacent faces have Neumann boundary conditions (ie.,~there are no Neumann-Neumann corners or edges). Our regularity and well-posedness results are formulated in a scale of weighted Sobolev spaces \(K^{m+1}_{ a+1} (D)\) of Kondrat'ev type in \(D\). We prove that the parametric, elliptic PDEs \((P_y)_{y \in U}\) admit a shift theorem which is uniform in the parameter sequence \(y \in U\). Specifically, if coefficients \(a^{ij}_{pq} (x, y)\) depend on the parameter sequence \(y = (y_k)_{k \ge 1}\) in an affine fashion, ie.~\(a^{ij}_{pq} = a^{\overline{ij}}_{pq0} + \sum_{k \ge 1} y_k \psi^ {ij}_{pqk}\), and if the sequences \(\|\psi^{ij}_{pqk}\|_{W^{m,\infty}(D)}\) are \(p\)-summable for some \(0 < p < 1\), then the parametric solution \(u_y\) admit an expansion into tensorized Legendre polynomials \(L_\nu (y)\) such that the corresponding coefficient sequence \({\bf u} = (u_\nu) \in \ell^p ( {\cal F}; {\cal K}^{m+1}_{a+1} (D))\). Here, we denote by \({\cal F} \subset IN^{IN}_0\) the set of sequences \(\{k_n\}_{n\in IN}\) with \(k_n \in IN_0\) with only finitely many non-zero terms, and by \(Y = \ell^\infty(IN)\) and \(U = B_1 (Y)\), the open unit ball of \(Y\). We identify the parametric solution u with its coefficient vector \({\bf u} = (u_\nu)_{\nu \in {\cal F}}, u_\nu \in V\), in the ''polynomial chaos'' expansion with respect to tensorized Legendre polynomials on \(U\). We also show quasioptimal algebraic orders of convergence for Finite Element approximations of the parametric solutions \(u(y)\) from suitable Finite Element spaces in two and three dimensions. Let \(t = m/d\) and \(s = 1/p - 1/2\) for some \(p \in (0, 1]\) such that {\bf u} \(= (u_\nu) \in \ell^p({\cal F}; {\cal K}^{m+1}_{a+1}(D))\). We then show that, for each \(m \in IN\), exists a sequence \(\{S_\ell\}_{\ell \ge 0}\) of nested, finite dimensional spaces \(S_\ell \subset L ^2 (U, \mu; V)\) such that \(M_\ell = {\rm dim}(S_\ell) \rightarrow \infty\) and such that the Galerkin projections \(u_\ell \in S_\ell\) of the solution \(u\) onto \(S_\ell\) satisfy \begin{equation*} \|u - u_\ell\|_{L^2 (U,\mu;V)} \le C\, {\rm dim}(S_\ell)^{- \min {s,t}} \|f\|_{H^{m -1} (D)} . \end{equation*} The sequence \(S_\ell\) is constructed using a nested sequence \(V_\mu \subset V\) of Finite Element space in \(D\) with graded mesh refinements toward the singular boundary points of the domain \(D\) as in [7, 9, 27]. Our sequence \(V_\mu\) is independent of \(y\). Each subspace \(S_\ell\) is then defined by a finite subset \(\Lambda_\ell \subset {\cal F}\) of ''active polynomial chaos'' coefficients \(u_\nu \in V\), \(\nu \in \Lambda_\ell\) in the Legendre chaos expansion of \(u\) which, in turn, are approximated by \(u_\nu \in V_{\mu(\ell,\nu)}\) for each \(\nu \in \Lambda_\ell\), with a suitable choice of \(\mu(\ell,\nu)\).

Keywords:

@Techreport{NS12_464,
  author = {V. Nistor and Ch. Schwab},
  title = {High order Galerkin approximations for parametric second order elliptic partial differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-21.pdf },
  year = {2012}
}

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