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.


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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