Convergence and quasi-optimality of an adaptive finite element method for semilinear elliptic problems on L² errors

Abstract

In this paper, we prove the convergence and quasi-optimality of an adaptive finite element method for semilinear elliptic problems on $L^2$ errors by keeping sufficiently mildly graded meshes. Additional refinements are made to keep the meshes sufficiently mildly graded, but we find that it does not compromise the quasi-optimality of the adaptive finite element method presented in this paper. Numerical examples are provided to illustrate our theoretical results.

Keywords:

• Adaptive finite element method
• convergence
• quasi-optimality
• semilinear elliptic problems
• $L^2$ errors

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N30
• 65N15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by Nanhu Scholars Program for Young Scholars of XYNU (2016), National Natural Science Foundation of China (11601466), and Doctoral Scientific Research Startup Fund of Xinyang Normal University (15021).