A Study Of Extended Hybridizable Discontinuous Galerkin Method For Interface Problem

In this paper, we present and analyse an unfitted mesh method for the interface problems. First, we provide the basic idea of the algorithm by using the Poisson interface problem as an example. By constructing a novel piecewise polynomial function in the vicinity of the interface, we are able to derive an extended Pois-son problem whose interface fits a given quasi-uniform triangular mesh exactly. Then we adopt a hybridizable discontinuous Galerkin (HDG) method to solve the extended problem with an appropriate choice of flux. Thus the jump condition-s are naturally introduced into the numerical schemes. In contrast to the existing approaches, the ansatz function is designed through a delicate piecewise quadratic Hermite polynomial interpolation with a post-processing via a standard Lagrange polynomial interpolation. Such an explicit function is able to accurately capture the jump conditions, and is proved to be unique, and offers a third-order approximation to the singular part of the underlying solution. More importantly, it is independent of the shape and the position of the interface. In addition, it leads to the high regu-larity of the solution of the extended interface problem to ensure that we can solve the problem with high accuracy solution by HDG method. Moreover, we provide rigorous error analysis to show that the scheme can achieve a second-order conver-gence rate for the approximation of the solution and its gradient. Ample numerical examples with complex interfaces demonstrate the expected convergence order and robustness of the method.Based on the successful experience for solving the Poisson interface problem, we study the parabolic interface problem, and mainly focus on the moving interface. Because the position and the shape of the interface change with time, we need to construct a high-accuracy piecewise polynomial to approximate the singular part of the exact solution at each time step, and use it to convert the original problem into a interface problem whose interface coincides with a given uniform mesh. Then HDG methods are used to discretize the spatial domain for the extended interface problem with an appropriate choice of flux, and the jump conditions are also naturally intro-duced into the numerical schemes. Consequently the accuracy of the second order convergence of the discretization is guaranteed. In order to ensure the numerical stability of the fully discrete scheme, the classical backward Euler scheme is used to discrete the time domain. It is worth pointing out that the idea of constructing the piecewise polynomial to approximate the singular part of the exact solution is the same at each time step, with only the position of the interface and the jump conditions changed. Fortunately, this change just have an impact on the right hand side of the final linear systems, and the coefficient matrix of the linear system will not change at all. Therefore, we only need to calculate and assemble the coefficient matrix of the linear system at the first time step, and then use the assembled coeffi-cient matrix repeatedly in the following steps. Therefore, it can greatly improve the efficiency of the algorithm. A large number of numerical experiments show that the method is not only stable, but also can ensure that the solution and its gradient in the L? norm with the second order convergent rate under the Cartesian grid.For the convenience of theoretical analysis, we only investigate the interface problems with affine jump conditions of the form [▽w · n] in the study of Poisson’s interface. In order to deal with more general interface problems with discontinuous coefficients, an iterative technique is introduced in this paper. By the iterative tech-nique, the solution of the general interface problem with discontinuous coefficients can be approximated by the solution of a series interface problems with a simple affine jump conditions. The iterative method is convergent if we choose the appro-priate convergence factor. Therefore, in order to get the high accuracy solution of the general interface problem, we only need to use the proposed numerical meth-ods to solve a series interface problem. Similar to the moving interface problem, there need to construct a piecewise polynomial to approximate the singular part of the solution according to the jump condition. In order to verify the effectiveness of the algorithm, we investigate the Helmholtz interface problem with a first-order absorbing boundary condition on the torus region. Numerical experiments show that the numerical method is not only stable, but also has second order convergence accuracy in the sense of L2 norm for the solution and its gradient.