劉 雪峰

For Poisson's equation over a polygonal domain of general shape, the solution of which may have a singularity around re-entrant corners, we provide an explicit a priori error estimate for the approximate solution obtained by finite element methods of high degree. The method used herein is a direct extension of the one developed in preceding paper of the second and third listed authors, which provided a new approach to deal with the singularity by using linear finite elements. In the present paper, we also give a detailed discussion of the dependency of the convergence order on solution singularities, mesh sizes and degrees of the finite element method used.

In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the Newton-Kantorovich theorem, our method can prove the existence and local uniqueness of the solution in the neighborhood of its approximation. In the treatment of polygonal domains with an arbitrary shape, which gives a singularity of the solution around the re-entrant corner, the computable error estimate of a projection into the finite-dimensional function space plays an essential role. In particular, the lack of smoothness of the solution makes classical error estimates fail on nonconvex domains. By using the Hyper-circle equation, an alternative error estimate of the projection has been proposed. Additionally, a new residual evaluation method based on the mixed finite element method works well. It yields more accurate evaluation than the existing method. The efficiency of our method is shown through illustrative numerical results on several polygonal domains.

有限要素法の誤差評価には、様々な誤差定数が現れる。その具体的な値は決定しにくい事もあり、存在のみが論じられることが多かったが、精度保証、事後誤差評価、適応型計算など定量的な誤差評価の重要性が増してきた結果、より具体的な値が必要になっている。本研究では、適合および非適合 1 次三角形有限要素法で現れる誤差定数を解析して、有限要素解の定量的な事前誤差評価と事後誤差評価を得た。