Extraction immersed finite element method for Stokes flow with singular forces

Abstract

We propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface. By considering a suitable function which has the same jumps as the solution, we transform the problem into the one without jumps. Then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in [1,2,3]. Some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested. Our method can also handle the case when the interface passes through grid points. We believe the treatment of such cases was first resolved in our paper. Numerical experiments for various problems show that second order convergence in $\It{L^2}$ and first order in $\It{H^1}$-norms. Moreover, the convergence order is very robust for all problems tested. Next, we consider a stationary, constant viscosity, incompressible Stokes flow with singular forces along one or several interfaces. Assuming only the jumps of the pressure are present along the interface, we develop a new numerical scheme for such a problem. By constructing an approximate singular function and removing it, we can apply a standard finite element method to solve it. A main advantage of our scheme is that one can use a uniform grid. We observe optimal $\It{O(h)}$ order for the pressure and $\It{O(h^2)}$ order for the velocity.