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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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