A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge co...

We present some new a priori estimates of the solutions to the second-order elliptic and parabolic interface problems. The novelty of these estimates lies in the explicit appearance of the discontinuous coefficients and the jumps of coefficients across the interface.

We define a new finite element method for a steady state elliptic problem with discontinuous diffusion coefficients where the meshes are not aligned with the interface. We prove optimal error estimates in the L norm and H weighted semi-norm independent of the contrast between the coefficients. Numerical experiments validating our theoretical findings are provided.

In this paper we introduce a discontinuous Galerkin method on polygonal meshes. This method arises from the Discontinuous Galerkin Composite Finite Element Method (DGFEM) for source problems on domains with micro-structures. In the context of the present paper, the flexibility of DGFEM is applied to handle polygonal meshes. We prove the a priori convergence of the method for both eigenvalues an...

In this paper, weak approximations of multi-dimensional stochastic differential equations with discontinuous drift coefficients are considered. Here as the approximated process, the Euler-Maruyama approximation of SDEs with approximated drift coefficients is used, and we provide a rate of weak convergence of them. Finally we present a rate of weak convergence of the Euler-Maruyama approximation...

This paper reviews some recent numerical methods for hyperbolic equations with singular (discontinuous or measure-valued) coefficients. Such problems arise in wave propagation through interfaces or barriers, or nonlinear waves through singular geometries. The connection between the well-balanced schemes for shallow-water equations with discontinuous bottom topography and the Hamiltonian preserv...

We propose new discontinuous finite element methods that can be applied to one-dimensional elliptic problems with discontinuous coefficients. These methods are based on a class of higher degree immersed finite element spaces and can be used with a mesh independent of the location of coefficient discontinuity. Numerical experiments are presented to show that these methods can achieve optimal con...

This paper is devoted to the study of some nonlinear hyperbolic equations or systems with discontinuous coefficients or with source terms. The common feature of the considered problems is the fact that the jacobian matrix of an associated autonomous system is not diagonalizable in R for many values of the unknown (leading to linear ill posed problems). However, the nonlinear problems appear to ...

We study the Cauchy problem for an equation with first order partial derivatives and two independent variables. One of coefficients is discontinuous. Hence characteristics are piecewise smooth curves solution to (understood in some generalized sense) has specific properties. In particular, solutions not defined domain discontinuous unextendable other domain.