A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations

✍ Scribed by G. Goodsell; J. R. Whiteman

• Publisher: John Wiley and Sons
• Year: 1989
• Tongue: English
• Weight: 523 KB
• Volume: 27
• Category: Article
• ISSN: 0029-5981
• DOI: 10.1002/nme.1620270304

✦ Synopsis

Piecewise linear finite element approximations to two-dimensional Poisson problems are treated. For simplicity, consideration is restricted to problems having Dirichlet boundary conditions and defined on rectangular domains R which are partitioned by a uniform triangular mesh. It is also required that the solutions u E H3(R). A method is proposed for recouering the gradients of the finite element approximations to a root mean square accuracy of O(h2), both at element edge mid-points and element vertices, using simple averaging schemes over adjacent elements. Piecewise linear interpolants (respectively discontinuous and continuous) are then fitted to these recovered gradients, and are shown to be. O ( h 2 ) estimates for Vu in the la,-norm, and thus superconcergent. A discussion is given of the extension of the results to problems with more general region and mesh geometries, boundary conditions and with solutions of lower regularity, and also to other second-order elliptic boundary value problems, e.g. the problem of planar linear elasticity.