A weak Galerkin mixed finite element method for biharmonic equations. (English) Zbl 1283.65109
Iliev, Oleg P. (ed.) et al., Numerical solution of partial differential equations: Theory, algorithms, and their applications. Papers based on the international symposium, Sozopol, Bulgaria, June 7–8, 2013. In Honor of Professor Raytcho Lazarov’s 40 years of research in computational methods and applied mathematics and on the occasion of his 70th birthday. New York, NY: Springer (ISBN 978-1-4614-7171-4/hbk; 978-1-4614-7172-1/ebook). Springer Proceedings in Mathematics & Statistics 45, 247-277 (2013).
The weak Galerkin method, first introduced by two of the authors, J. Wang and X. Ye [J. Comput. Appl. Math. 241, 103–115 (2013; Zbl 1261.65121)] for second-order elliptic problems, is applied to the Ciarlet-Raviart mixed formulation for the biharmonic equation. The weak Galerkin method is an extension of the standard Galerkin finite element method, where derivatives are substituted by weakly defined derivatives of functions with discontinuity. For second-order elliptic problems, a stabilization of the method works for finite element partitions of arbitrary polygonal or polyhedral type. The authors want to demonstrate the portability of the method to the biharmonic problem. For the 2d-case and triangular, rectangular finite element meshes they provide a detailed error analysis, which is based on mesh dependent norms. Some computational results confirm the established theory.
For the entire collection see [Zbl 1269.65002].
For the entire collection see [Zbl 1269.65002].
Reviewer: Gunther Schmidt (Berlin)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 65N15 | Error bounds for boundary value problems involving PDEs |
