A p-TYPE SUPERCONVERGENT RECOVERY METHOD FOR FE ANALYSIS WITH LAGRANGE ELEMENTS ON TWO-DIMENSIONAL POISSON EQUATIONS

A p-type superconvergent recovery method for the finite element analysis with Lagrange elements on two-dimensional Poisson equations is proposed. Based on the superconvergent properties of mesh nodal displacements in finite element solutions, the method is inspired by the idea of dimensionality reduction of two-dimensional problems by the finite element method of lines. A row of adjacent elements with common opposite edges is taken out as a sub-domain. A local boundary value problem of the original partial differential equations on it which the true solution approximately satisfies is established by setting finite element solutions on each element's other opposite edges as Dirichlet boundary conditions. By increasing the element order along the elements' opposite edges direction unidirectionally, the local boundary value problem is solved by the finite element method to obtain the superconvergent displacement solution on the opposite edges of each element in this sub-domain. The superconvergent solution on the other opposite edges of elements can be obtained similarly with another sub-domain with respect to the edges to be recovered. Based on the recovered edge solutions, each element domain is taken out. The original Poisson equation on it is solved using a higher order Lagrange element with the superconvergent solution on its edges set as Dirichlet boundary conditions. Thus, the superconvergent solution of the whole domain can be obtained. Numerical examples show that the method can significantly improve the accuracy and convergence order of solutions with a small amount of computation.