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Optimal estimates for lower and upper bounds of approximation errors in the $$p$$-version of the finite element method in two dimensions. (English) Zbl 0970.65117
To increase the accuracy of finite element approximations, we can refine the mesh ($$h$$-version), increase the element degree $$p$$ ($$p$$-version), or combine both methods ($$h$$-$$p$$-version). The $$p$$-version converges in the energy norm at least as fast as the $$h$$-version for the approximation of regular problems and it is more performant than the $$h$$-version when singularities occur, for instance of $$r^\gamma$$-type near vertices of nonsmooth domains.
This paper, by eminent specialists of such problems, introduces the best mathematical tool for the $$p$$-version in two and three dimensions, i.e., the weigthed Besov spaces with Jacobi weights. Then, by using these spaces, they characterize the singularity of the solution of problems on nonsmooth domains, they precisely estimate the lower and upper bounds of the approximation error and they establish optimal convergence of the $$p$$-version, mainly in two dimensions.

##### MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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