Convergence and quasi-optimality of an adaptive continuous interior multi-penalty finite element method

Abstract

An adaptive continuous interior multi-penalty finite element method (ACMPFEM) for symmetric second-order linear elliptic equations is considered. The proposed ACMPFEM is defined using a bilinear form which not only penalize the jumps of the first order normal derivatives across the element edges but also the jumps of all normal derivatives up to order p. In addition, compared with the analyses for adaptive finite element method, extra works are done to overcome the difficulties caused by the additional penalty terms. To be precise, the first difficulty is that the variational formulation of the ACMPFEM is not consistent with the continuous problem for weak solution u just in $H^1\left(\mathrm{\Omega }\right)$ and the second difficulty lies in the proof of the optimality of our ACMPFEM. Furthermore, to prove the convergence and quasi-optimality of the ACMPFEM, an approximate Galerkin orthogonality in which the penalty term is regarded as a perturbation, and estimating the perturbation very carefully is used and the definition of our approximation class has to be involved with the penalty terms. Numerical tests are provided to verify the theoretical results and show advantages of the ACMPFEM.

Keywords:

• Continuous interior multi-penalty finite element method
• adaptive algorithm
• convergence
• quasi-optimality

AMS subject classifications:

• 65N12
• 65N15
• 65N30
• 78A40

Acknowledgments

This work was partially supported by the NSFC (No. 11401272) and by the Natural Science Foundation of Jiangsu Province of China (No. BK20140105).

Availability of data and materials

The authors declare that all data and material in the article are available and veritable.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially the National Natural Science Foundation of China [grant number 11401272] and the Natural Science Foundation of Jiangsu Province of China [grant number BK20140105].