Weak Galerkin finite element method for second order problems on curvilinear polytopal meshes with Lipschitz continuous edges or faces

In this paper, the weak Galerkin finite element method for second order problems on curvilinear polytopal meshes with Lipschitz continuous edges or faces was analyzed. The method here is designed to deal with second order problems with complex boundary conditions or complex interfaces. With Lipschitz continuous boundary or interface, the method's optimal convergence rate for $H^1$ and $L^2$ error estimates were obtained. Arbitrary high orders can be achieved.Comment: arXiv admin note: substantial text overlap with arXiv:1805.0092

The weak Galerkin finite element method for Stokes interface problems with curved interface

In this paper, we develop a new weak Galerkin finite element scheme for the Stokes interface problem with curved interfaces. We take a unique vector-valued function at the interface and reflect the interface condition in the variational problem. Theoretical analysis and numerical experiments show that the errors can reach the optimal convergence order under the energy norm and $L^2$ norm

Residual-based a posteriori error estimation for immersed finite element methods

In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and efficient with reliability constant independent of the location of the interface. Numerical results indicate that the efficiency constant is independent of the interface location and that the error estimation is robust with respect to the coefficient contrast