Ultraconvergence of high order FEMs for elliptic problems with variable coefficients

• Published in: Numerische Mathematik Vol. 136; no. 1; pp. 215 - 248
• Main Authors: He, Wen-ming, Zhang, Zhimin, Zou, Qingsong
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2017; Springer Nature B.V
• Subjects: Exact solutions; Finite element method; Green's functions; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Simulation; Theoretical; 65N25; 65N15; 65N30
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-016-0838-6

In this paper, we investigate local ultraconvergence properties of the high-order finite element method (FEM) for second order elliptic problems with variable coefficients. Under suitable regularity and mesh conditions, we show that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the post-precessed k th ( k ≥ 2 ) order finite element solution converges to the gradient of the exact solution with order O ( h k + 2 ( ln h ) 3 ) . The proof of this ultraconvergence property depends on a new interpolating operator, some new estimates for the discrete Green’s function, a symmetry theory derived in [ 26 ], and the Richardson extrapolation technique in [ 20 ]. Numerical experiments are performed to demonstrate our theoretical findings.