Higher-order finite element methods for elliptic problems with interfaces

We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.Comment: 26 pages, 6 figures. An earlier version of this paper appeared on November 13, 2014 in http://www.brown.edu/research/projects/scientific-computing/reports/201

Singular function mortar finite element methods

This is the published version, also available here: http://dx.doi.org/10.2478/cmam-2003-0014.We consider the Poisson equation with Dirichlet boundary conditions on a polygonal domain with one reentrant corner. We introduce new nonconforming finite element discretizations based on mortar techniques and singular functions. The main idea introduced in this paper is the replacement of cut-off functions by mortar element techniques on the boundary of the domain. As advantages, the new discretizations do not require costly numerical integrations and have smaller a priori error estimates and condition numbers. Based on such an approach, we prove optimal accuracy error bounds for the discrete solution. Based on such techniques, we also derive new extraction formulas for the stress intensive factor. We establish optimal accuracy for the computed stress intensive factor. Numerical examples are presented to support our theory