On Some Convergence Properties for Finite Element Approximations to the Inverse of Linear Elliptic Operators

Keywords: linear elliptic problems, finite element approximation, norm estimation of the inverse operator, convergence theorem

Abstract

This paper deals with convergence theorems of the Galerkin finite element approximation for the second-order elliptic boundary value problems. Under some quite general settings, we show not only the pointwise convergence but also prove that the norm of approximate operator converges to the corresponding norm for the inverse of a linear elliptic operator. Since the approximate norm estimates of linearized inverse operator play an essential role in the numerical verification method of solutions for non-linear elliptic problems, our result is also important in terms of guaranteeing its validity. Furthermore, the present method can also be applied to more general elliptic problems, e.g., biharmonic problems and so on.

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Published
2022-09-02
How to Cite
Kinoshita, T., Watanabe, Y., & T. Nakao, M. (2022). On Some Convergence Properties for Finite Element Approximations to the Inverse of Linear Elliptic Operators. Acta Cybernetica, 26(1), 71-82. https://doi.org/10.14232/actacyb.294906
Section
Special Issue of SCAN 2020