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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.

##### MSC:
 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
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