A Note on Improving the Rate of Convergence of ‘High Order Finite Elements’ on Polygons
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چکیده
Let u and uVn be the solution and, respectively, the finite element solution of the Poisson’s equation ∆u = f with zero boundary conditions. We construct for any m ∈ N and any polygon P a sequence of finite dimensional subspaces Vn such that ‖u−uVn‖H1 ≤ C dim(Vn)‖f‖Hm−1 , where f ∈ Hm−1(P) is arbitrary and C is a constant that depends only on P (we do not assume u ∈ Hm+1(P)). Although the final result is in terms of the “usual” Sobolev spaces, the proof relies on estimates for the Poisson problem in Sobolev spaces with weights. Other “hm”–type approximation results are also obtained. This is an announcement, but some sketches of the proofs of the main results are provided. Full details of the proofs and complete references will be provided in a different paper.
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A Note on Improving the Rate of Convergence of ` High Order Finite Elements ' on Polygonsconstantin
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تاریخ انتشار 2003