## 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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