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On \(n\)-widths for elliptic problems. (English) Zbl 0963.35047

The widths of solution sets of elliptic boundary value problems having the form \[ Lu:= -\nabla(A(x)\nabla u)+ c(x)u= f\quad\text{on }\Omega\subset \mathbb{R}^d,\tag{1} \] \(u= 0\) on \(\partial\Omega\) are considered, where the matrix \(A= (a_{ij})^d_{i,j=1}\) with \(a_{ij}\in L^\infty(\Omega)\) is assumed to be symmetric positive definite and \(c\in L^\infty(\Omega)\), \(c\geq 0\).
For a normed linear space \(X\) and a subset \(A\subset X\) the \(n\)-width is defined by \[ d_n(A, X)= \inf_{\varepsilon_n} \sup_{f\in A} \inf_{g\in E_n} \|f-g\|_X, \] where the first infimum is taken over all subspaces \(E_n\) of \(X\) of dimension \(n\in\mathbb{N}\).
There are considered two types of sets \(A\), namely, solution sets \(A^S\) of (1) for right-hand sides \(f\) with finite regularity and solution set \(A^G\) for analytic right-hand sides. Sharp upper and lower bounds for \(d_n(A^S,X)\) and \(d_n(A^G,X)\) are obtained and it is shown that \(d_n(A,X)\) depends only on the regularity of the right-hand side \(f\), the upper and lower bounds on the eigenvalues of \(A\), and the upper and lower bounds on the coefficient \(c\).
The author studies the following two cases: the case of rough coefficients (\(A\) and \(c\) are in \(L^\infty(\Omega)\) but the “energy norm” is equivalent to the usual \(H(\Omega)\) norm) and the case of singularity perturbed equations of elliptic-elliptic type (the eigenvalues of \(A\) are small compared with \(c\)).
In the first case the \(n\)-widths \(d_n(A^S,X)\), \(d_n(A^G,X)\) are the same as in the case of smooth coefficients. In the second case for \(d_n(A^S,X)\) the \(n\)-width deteriorates pre-asymptotically when the size of the eigenvalues of \(A\) tends to zero, and for \(d_n(A^G,X)\) the \(n\)-width decreases exponentially independent of the size of the eigenvalues of \(A\).
In the end of this paper an application to the finite element method is given.

MSC:

35J25 Boundary value problems for second-order elliptic equations
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
35R05 PDEs with low regular coefficients and/or low regular data
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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