Related papers: $C^0$ discontinuous Galerkin finite element method…
We introduce a new weak Galerkin finite element method whose weak functions on interior neighboring edges are double-valued for parabolic problems. Based on $(P_k(T), P_{k}(e), RT_k(T))$ element, a fully discrete approach is formulated with…
A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity-pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general…
We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in $\mathbb{R}^3$. A priori error estimates, taking both the approximation of the surface and the…
This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution…
We present a numerical method to model the dynamics of inextensible biomembranes in a quasi-Newtonian incompressible flow, which better describes hemorheology in the small vasculature. We consider a level set model for the fluid-membrane…
We consider a system of second order non-linear elliptic partial differential equations that models the equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin finite element…
A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different…
This paper develops and analyzes an interior penalty discontinuous Galerkin (IPDG) method using piecewise linear polynomials for the indefinite time harmonic Maxwell equations with the impedance boundary condition in the three dimensional…
Lower-order $P_2$ finite elements are popular for solving fourth-order elliptic PDEs when the solution has limited regularity. A priori and a posteriori error estimates for von Karman equations are considered in Carstensen et al. (2019,…
We study fully discrete linearized Galerkin finite element approximations to a nonlinear gradient flow, applications of which can be found in many areas. Due to the strong nonlinearity of the equation, existing analyses for implicit schemes…
We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such…
A new modified Galerkin / Finite Element Method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low-order Lagrange finite element spaces, despite the…
This paper is devoted to a weak Galerkin (WG) finite element method for linear poroelasticity problems where weakly defined divergence and gradient operators over discontinuous functions are introduced. We establish both the continuous and…
The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method builds upon the formulation introduced in Bertalmio et al., J. Comput. Phys., 174 (2001),…
We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the…
The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous…
A new primal-dual weak Galerkin (PD-WG) finite element method was developed and analyzed in this article for first-order linear convection equations in non-divergence form. The PD-WG method results in a symmetric discrete system involving…
We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori…
A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at…
A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in \cite{wwnondiv}, the system of equations…