Optimal quadratic element on rectangular grids for $$H^1$$ H 1 problems
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Optimal quadratic element on rectangular grids for H 1 problems Huilan Zeng1 · Chen-Song Zhang1 · Shuo Zhang1 Received: 8 March 2019 / Accepted: 13 June 2020 © Springer Nature B.V. 2020
Abstract In this paper, a piecewise quadratic finite element method on rectangular grids for H 1 problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is proved to be O(h 2 ) in the energy norm on uniform grids over a convex domain. A lower bound of the L 2 -norm error is also proved, which makes the capacity of this scheme more clear. For the eigenvalue problem, the computed eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented to verify the theoretical findings. Keywords Optimal quadratic element · Rectangular grids · Boundary value problem · Eigenvalue problem · Lower bound Mathematics Subject Classification 65N15 · 65N22 · 65N25 · 65N30
1 Introduction The design and capacity analysis of the discretization schemes for the source problem (say, the boundary value problem) and the eigenvalue problem are key issues in numerical analysis and approximation theory. When the approximation of functions
Communicated by Axel Målqvist.
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Shuo Zhang [email protected] Huilan Zeng [email protected] Chen-Song Zhang [email protected]
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LSEC, ICMSEC and NCMIS, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
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H. Zeng et al.
in Sobolev spaces is performed using piecewise polynomials defined on a domain partition, lower-degree polynomials are often preferred in order to yield a simpler interior structure. A finite element scheme with polynomials of the total degree no more than k, denoted by Pk , is called optimal if it achieves O(h k+1−m ) accuracy in the energy norm for H m elliptic problems. In this paper, we present an optimal quadratic element scheme for H 1 problems, including the source problem and the eigenvalue problem, on rectangular grids, and present its error analysis. The study of optimal finite element schemes has been attracting wide interests for several decades. For the case wherein the grid comprises simplexes, there are already some systematic results. It is known that the Lagrange finite elements of arbitrary degree on domains of arbitrary dimension are optimal conforming elements for secondorder elliptic problems. At the same time, a family of minimal-degree nonconforming finite elements, known as Wang-Xu or Morley-Wang-Xu family, is proposed by [28], where mth degree polynomials are empolyed for 2mth order elliptic problems in Rn for any n ≥ m. The generalisation to the cases where n < m is attracting increasing research interest (see, e.g., [30]). These spaces can be naturally used for both the source problem and the eigenvalue problem. On the other hand, to clarify the capacity of the schemes clearly, some kinds of extremal analysis have also been conducted, including, e.g., lower bounds of the
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