zbMATH — the first resource for mathematics

A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. (English) Zbl 1014.65115
The authors consider an elliptic system parameterized by a scalar \(\mu\in[0,\mu_0]\) of the form \[ a_0(u(\mu),v)+\mu a_1(u(\mu),v)=f(v)\qquad \forall v\in Y\tag{*} \] where \(Y\) is an appropriate function space, \(a_0\) and \(a_1\) are continuous and symmetric, \(a_0\) is coercive and \(a_1\) is postitive semi-definite. For each choice of \(\mu\) it is possible to approximate \(u(\mu)\) to arbitrary accuracy by a member \(u^{\mathcal N}(\mu)\) of an approximating subspace of \(Y^{\mathcal N}\subset Y\) of sufficiently large but finite dimension \(\mathcal N\).
The authors prove that it is possible to choose \(N\ll {\mathcal N}\) sample values \(\mu_n\), \(n=1,2,\ldots,N\), logarithmetically distributed in the interval \([0,\mu_0]\), with the following property. For each \(\mu_n\), denote an approximate solution to \((*)\) in \(Y^{\mathcal N}\) by \(u^{\mathcal N}_n\), and denote the span of these approximates as \(W^{\mathcal N}_N\). If \(N\) is larger than a critical value \(N_0\), then an approximation to \(u(\mu)\) can be found in \(W^{\mathcal N}_N\) so that \[ |||u(\mu)-u^{\mathcal N}_N(\mu)|||\leq |||u(\mu)-u^{\mathcal N}(\mu)|||+C|||u(0)|||e^{-N/N_0} \] where \(C\) denotes a constant depending only on \(a_0\), \(a_1\), and \(\mu_0\), and \(|||\cdot|||\) denotes the norm induced by \(a_0(u,v)\).
Numerical testing indicates that the logarithmic distribution is optimal and that a similar result might hold in more than one dimension.

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI