A superconvergent nonconforming quadrilateral spline element for biharmonic equation using the B-net method

• Published in: Computational & applied mathematics Vol. 39; no. 2
• Main Authors: Li, Chong-Jun, Jia, Yan-Mei
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.05.2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied physics; Biharmonic equations; Bivariate analysis; Computational mathematics; Computational Mathematics and Numerical Analysis; Degrees of freedom; Integrals; Interpolation; Mathematical analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Quadrilaterals; Shape functions; Spline interpolation bases; Quadrilateral element; Superconvergence; The B-net method; Biharmonic problem; 65D07; 65N30
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-020-1105-0

In this paper, we construct a new nonconforming quadrilateral element with 12 degrees of freedom to solve the biharmonic problems. T h is a triangulated quadrangulation of the domain Ω . For a quadrilateral element Q T , the finite element space, which contains P 3 ( Q T ) , is a subspace of the bivariate spline space S 3 1 ( Q T ) . The degrees of freedom are chosen as the four point values, the four edge integrals of the shape functions and the edge integrals of their normal derivatives such that the weak continuity between elements can be satisfied. Accordingly, we explicitly establish 12 spline interpolation bases in the B-net form. Error estimates are given with optimal convergence order in both discrete H 2 and H 1 seminorms. The proposed element NCQS12 can get the superconvergence results with theoretical proof when T h is an uniform parallelogram mesh. Some degenerate meshes are considered subsequently. Finally, we do some numerical experiments to verify the theoretical analysis. Numerical results show that the proposed element performs well over the asymptotically regular parallelogram meshes, which is same as over the uniform parallelogram meshes.