Hybridizable discontinuous Galerkin method (HDG) for Stokes interface flow

In this paper, we present a hybridizable discontinuous Galerkin (HDG) method for solving the Stokes interface problems with discontinuous viscosity and variable surface tension. The jump condition of the stress tensor across the interface is naturally incorporated into the HDG formulation through a constraint on the numerical flux. The most important feature of HDG method compared to other DG methods is that it reduces the number of globally coupled unknowns significantly when high order approximate polynomials are used. For problems with polygonal interfaces, it provides optimal convergence rates of order k+1k+1 in L2L2-norm for the velocity, pressure and as well as the gradient of velocity. Furthermore, a new approximate velocity can be obtained by an element-by-element postprocessing which converges with order k+2k+2 in the L2L2-norm. For Stokes interface problems with curved interfaces, we use general curvilinear element to ensure the optimal convergence rates. An error estimate is given for the approximation of the interface. It indicates that curvilinear elements of degree 2k+12k+1 should be used for optimal convergence rate of order k+1k+1.