A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations.Distinguished from the Runge-Kutta discontinuous Galerkin method(RKDG)and the finite element time domain method(FETD),in our scheme,discontinuous Galerkinmethods are used to discretize not only the spatial domain but also the temporal domain.The proposed numerical scheme is proved to be unconditionally stable,and a convergent rate O((△t)^(r+1)+h^(k+1/2))is established under the L^(2)-normwhen polynomials of degree atmost r and k are used for temporal and spatial approximation,respectively.Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction.An ultra-convergence of order(△t)^(2r+1) in time step is observed numerically for the numerical fluxes w.r.t.temporal variable at the grid points.
In this article, we introduce a coupled approach of local discontinuous Galerkin and standard finite element method for solving convection diffusion problems. The whole domain is divided into two disjoint subdomains. The discontinuous Galerkin method is adopted in the subdomain where the solution varies rapidly, while the standard finite element method is used in the other subdomain due to its lower computational cost. The stability and a priori error estimate are established. We prove that the coupled method has O((ε1 / 2 + h 1 / 2 )h k ) convergence rate in an associated norm, where ε is the diffusion coefficient, h is the mesh size and k is the degree of polynomial. The numerical results verify our theoretical results. Moreover, 2k-order superconvergence of the numerical traces at the nodes, and the optimal convergence of the errors under L 2 norm are observed numerically on the uniform mesh. The numerical results also indicate that the coupled method has the same convergence order and almost the same errors as the purely LDG method.
In this paper,we propose an accelerated search-extension method (ASEM) based on the interpolated coeffcient finite element method,the search-extension method (SEM) and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems,but also reduces the expensive computation greatly. The numerical results in 1-D and 2-D cases will show the effciency of our approach.
In this paper,we consider the local discontinuous Galerkin method (LDG) for solv-ing singularly perturbed convection-diffusion problems in one- and two-dimensional set-tings.The existence and uniqueness of the LDG solutions are verified.Numerical ex-periments demonstrate that it seems impossible to obtain uniform superconvergence fornumerical fluxes under uniform meshes.Thanks to the implementation of two-type dif-ferent anisotropic meshes,i.e.,the Shishkin and an improved grade meshes,the uniform2p+1-order superconvergence is observed numerically for both one-dimensional and two-dimensional cases.
In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the underlying problem. Unconditional L2-stability and error estimate of order O τr+1+ hk+1/2 are obtained when polynomials of degree at most r and k are used for the temporal discretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r + 1 in temporal variable t.
This work is to provide general spectral and pseudo-spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential equations.The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation.For some spectral and pseudo-spectral Jacobi-Petrov-Galerkin methods,a rigorous error analysis in both L2_(ω^(α,β))^(2),and L^(∞)norms is given provided that both the kernel function and the source function are sufficiently smooth.Numerical experiments validate the theoretical prediction.
