Mixed Direct Discontinuous Galerkin Method for the Biharmonic Equation

A new discontinuous Galerkin finite element method for solving diffusion problems is introduced. Unlike the traditional local discontinuous Galerkin method, the scheme called the direct discontinuous Galerkin (DDG) method is based on the direct weak formulation for solutions of parabolic equations in each computational cell and lets cells communicate via the numerical flux $\widehat{u_x}$ only. We propose a general numerical flux formula for the solution derivative, which is consistent and conservative; and we then introduce a concept of admissibility to identify a class of numerical fluxes so that the nonlinear stability for both one-dimensional (1D) and multidimensional problems are ensured. Furthermore, when applying the DDG scheme with admissible numerical flux to the 1D linear case, kth order accuracy in an energy norm is proven when using kth degree polynomials. The DDG method has the advantage of easier formulation and implementation and efficient computation of the solution. A series of numerical examples are presented to demonstrate the high order accuracy of the method. In particular, we study the numerical performance of the scheme with different admissible numerical fluxes.

A third‐order singularly perturbed problem of reaction–diffusion type is solved numerically by the local discontinuous Galerkin (LDG) method. By using the local Gauss–Radau projection for the primal variable u, but local L2 projection for the two auxiliary variables p and q, we obtain an optimal convergence rate of order (or up to a logarithmic factor) in the energy norm, uniformly in the singular perturbation parameter. Here is the degree of the piecewise polynomials used in the discrete space and N is the number of mesh intervals. The layer‐adapted meshes are used including a standard Shishkin mesh, a Bakhvalov–Shishkin mesh, and a Bakhvalov‐type mesh. Numerical experiments are given to show the sharpness of our theoretical results.

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k‐th and ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017

A discontinuous Galerkin least-squares method for div–curl systems

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

Locally discontinuous but globally continuous Galerkin methods for elliptic problems

Conforming virtual element approximations of the two-dimensional Stokes problem

HDG method for linear parabolic integro-Differential equations

A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive $${\epsilon}$$) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

We propose and analyze an unfitted finite element method of arbitrary order for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. We apply a non-symmetric bilinear form and the boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. The optimal convergence rate under the energy norm is derived, and \(O(h^{-2})\)-upper bounds of the condition numbers are shown for the final linear systems. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.

In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional cavity. Finite element methods based on the variational formulation derived in [23] are considered. Due to the lack of regularity ofer, the existence and uniqueness of the discrete solutions and their convergence are proved by using the concept of collectively compact operators. An optimal convergence rate in the energy norm is also established.

Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L2 norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L2 norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017

In this paper, we propose the local discontinuous Galerkin method based on the generalized alternating numerical flux for solving the one-dimensional second-order wave equation with the periodic boundary conditions. Introducing two auxiliary variables, the second-order equation is rewritten into the first-order equation systems. We prove the stability and energy conservation of this method. By virtue of the generalized Gauss–Radau projection, we can obtain the optimal convergence order in L2-norm of Ohk+1 with polynomial of degree k and grid size h. Numerical experiments are given to verify the theoretical results.

A hierarchical approach to the a posteriori error estimation of isogeometric Kirchhoff plates and Kirchhoff–Love shells