The h-p version of the finite element method in three dimensions
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In the framework of the Jacobi-weighted Besov and Sobolev spaces, we analyze the approximation to singular and smooth functions. We construct stable and compatible polynomial extensions from triangular and square faces to prisms, hexahedrons and pyramids, and introduce quasi Jacobi projection operators on individual elements, which is a combination of the Jacobi projection and the interpolation at vertices and on sides of elements. Applying these results we establish the convergence of the h-p version of the finite element method with quasi uniform meshes in three dimensions for elliptic problems with smooth solutions or singular solutions on polyhedral domains in three dimensions. The rate of convergence interms of h and p is proved to be the best.