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We consider a Dirichlet problems with degenerate coefficients and with degeneration in angular points of domain. For these problems in various weight Sobolev norms estimates of solutions are derived. Using theory of Kolmogorov's width, for any Galerkin approximation a lower estimates of the accuracy in energy norm are obtained.
It is known, that the solution of Dirichlet problem for the elliptic equation with degenerate coefficients has unlimited derivatives around of the points of degeneration. Therefore, standard finite element method with a piecewise polynomial basis is inefficient for approximation of such solutions. For considered problems we present an optimal finite element method founded on a multiplicative allocation of singularities. The obtained estimates of convergence rate of method confirm results of the numerical experiments.
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