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

Abstract

Let $D \subset \mathbb{R}^d, d=2,3$, be a bounded domain with piecewise smooth boundary $\partial D$ and let $U$ be an open subset of 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 $P_yu=f_y$ are functions of $(x,y) \in D$ x $U$. Under suitable regularity assumptions on these coefficients and on the source term $f$, we establish a regularity result for the solution $u: D$ x $U$ $\to$ \mathbb{R} of the parametric, elliptic boundary value problem $P_yu(x,y)$ = $f_y(x)$ = $f(x,y),x \in D,y \in U$, with mixed Dirichlet-Neumann boundary conditions. Let $\partial D$ = $\partial _dD \cup \partial _nD$ 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 $\partial _dD$ 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 $K_{a+1}^{m+1} (D)$ 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 $a_{pq}^{ij} (x,y)$ depend on the parameter sequence $y$ = $(y_k)_{k \geq 1}$ in an affine fashion, ie. $a_{pq}^{ij}$ = $a_{pq0}^{ \overline {ij}}$ + $ \sum_{k \geq 1} y_k \psi_{pqk}^{ij}$, and if the sequences $|| \psi_{pqk}^{ij}||W^{m, \infty} (D)$ are $p$-summable for $0<p<1$, then the parametric solution $u_y$ admit an expansion into tensorized Legendre polynomials $L_v(y)$ such that the corresponding coefficient sequence $u$ = $(u_v) \in l^p (\mathcal{F}; \mathcal{K}_{a+1}^{m+1} (D))$. Here, we denote by $\mathcal{F} \subset \mathbb{N}_0^{\mathbb{N}}$ the set of sequences {$k_n$}$_{n \in \mathbb{N}}$ with $k_n \in \mathbb{N}_0$ with only finitely many non-zero terms, and by $Y$ = $l^{\infty} (\mathbb{N})$ und $U$ = $B_1(Y)$ , the open unit ball of $Y$. We identify the parametric solution u with its coefficient vector $u$= $(u_v)$$_{v \in \mathcal{F}},$ $u_v$ $\in$ $V$, n the ''polynomial chaos'' expansion with respect to tensorized Legendre polynomials on $U$. We also show quasioptimal algebraic orders of convergence for Finite Element approximations of the parametric solutions $u(y)$ from suitable Finite Element spaces in two and three dimensions. Let $t$ = $m/d$ and $s$ = $1/p -1/2$ for some $p \in (0.1]$ such that $\mathbf{u}$ = $(u_v) \in$ $l^p$ $(\mathcal{F}; \mathcal{K}_{a+1}^{m+1} (D))$. We then show that, for each $ m \in \mathbb{N}$, exists a sequence {$S_l$}$_{l \geq 0}$ of nested, finite dimensional spaces $S_l \subset L^2(U,y;V)$ such that $M_l$ = dim($S_l$) $\rightarrow$ $\infty$ and such that the Galerkin projections $u_l \in S_l$ of the solution $u$ onto $S_l$ satisfy $$|| u - u_l||_{L^2 (U, \mu ; V)} \leq C \ dim(S_l)^{- min \{ s,t \} } ||f||_{H^{m-1}(D)}$$ The sequence $S_l$ 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 $S_l$ is then defined by a finite subset $\wedge _l \subset \mathcal{F}$ of ''active polynomial chaos'' coefficients $u_v \in V, v \in \wedge _l$ in the Legendre chaos expansion of $u$ which, in turn, are approximated by $u_v \in V{\mu (l,v)}$ for each $v \in \wedge _l$ , with a suitable choice of $\mu (l,v)$.