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A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM

Yuping Zeng1, Feng Wang2, Zhifeng Weng3, Hanzhang Hu1   

  1. 1. School of Mathematics, Jiaying University, Meizhou 514015, China;
    2. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China;
    3. Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
  • Received:2019-01-21 Revised:2019-06-03 Published:2021-10-15
  • Supported by:
    The authors thank the anonymous referees for their valuable comments and suggestions which helped to improve the quality of this article. The first author was supported by Guangdong Basic and Applied Basic Research Foundation (Grant Nos. 2018A030307024 and 2020A1515011032), and by National Natural Science Foundation of China (Grant No. 11526097). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11871272 and 11871281). The third author was supported by National Natural Science Foundation of China (Grant No. 11701197). The fourth author was supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2018A0303100016).

Yuping Zeng, Feng Wang, Zhifeng Weng, Hanzhang Hu. A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM[J]. Journal of Computational Mathematics, 2021, 39(5): 755-776.

In this paper, we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems. We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method, though they have essentially different bilinear forms. More precisely, we prove its reliability and efficiency for the actual error measured in the standard DG norm. We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution. Numerical results are presented to verify the theoretical analysis.

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