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Weak Galerkin finite element methods (WG-FEMs) for H (curl ; Ω) and H (curl , div ; Ω) -elliptic problems are investigated in this paper. The WG method as applied to curl-curl and grad-div problems uses two operators: discrete weak curl and discrete weak divergence, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on the arbitrary shape of polyhedra and, at the same time, is parameter-free. The optimal order of convergence is established for the WG approximations in discrete H 1 norm and L 2 norm. In fact, theoretical convergence analysis holds under low regularity requirements of the analytical solution. Results of numerical experiments that corroborate the theoretical results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L 2 (L 2) and L 2 (H 1) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O (h r + 1) in L 2 (L 2) norm and O (h r) in L 2 (H 1) norm when the exact solution u ∈ L 2 (0 , T ; H r + 1 (Ω)) ∩ H 1 (0 , T ; H r − 1 (Ω)) , for some r ≥ 1. Numerical experiments are reported for several test cases to justify our theoretical convergence results. [ABSTRACT FROM AUTHOR]

The paper concerns numerical study of non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L ∞ (L 2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for non-homogeneous inner structures. [ABSTRACT FROM AUTHOR]

In this article a fitted finite element method is proposed and analyzed for wave equation with discontinuous coefficients. Typical semidiscrete and an implicit fully discrete schemes are presented and analyzed. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L ∞ ( L 2 ) norm. The convergence analysis relies heavily on time reconstructions of continuous and discrete solutions, in conjunction with some known results on elliptic interface problems. Finally, a numerical experiment is presented to verify our theoretical result. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to study finite element methods and their convergence for hyperbolic interface problems. Both semidiscrete and fully discrete schemes are analyzed. Optimal a priori error estimates in the and norms are derived for a finite element discretization where interface triangles are assumed to be curved triangles instead of straight triangles. The interfaces and boundaries of the domains are assumed to be smooth for our purpose. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study the finite element methods for second-order semilinear elliptic and parabolic interface problems in two dimensional convex polygonal domains. Optimal order error estimate in the -norm is proved for the semilinear elliptic interface problem when the grid lines follow the actual interface. An extension to the semilinear parabolic interface problem is considered, and both semidiscrete and fully discrete schemes are discussed. The convergence of the semidiscrete solution to the exact solution is shown to be of order in the -norm. Further, a fully discrete scheme based on backward Euler method is analyzed and optimal energy-norm error estimate is established. The interface is assumed to be of arbitrary shape but is smooth. [Copyright &y& Elsevier]