Related papers: Comparison results of $P_2$-finite elements for fo…
We devise new variants of the following nonconforming finite element methods: DG methods of fixed arbitrary order for the Poisson problem, the Crouzeix-Raviart interior penalty method for linear elasticity, and the quadratic $C^0$ interior…
A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise…
In this paper, we derive a priori error estimates for a class of interior penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE) functions for a classic second-order elliptic interface problem. The error estimation…
We present a weak finite element method for elliptic problems in one space dimension. Our analysis shows that this method has more advantages than the known weak Galerkin method proposed for multi-dimensional problems, for example, it has…
A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semi-linear problems with trilinear non-linearity and a general source. A quasi-optimal smoother extends the source term to the…
An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in…
We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving eigenvalue problems associated with second…
A new immersed finite element (IFE) method is developed for second-order elliptic problems with discontinuous diffusion coefficient. The IFE space is constructed based on the rotated Q1 nonconforming finite elements with the integral-value…
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…
In this paper, we study the convergence of adaptive mixed interior penalty discontinuous Galerkin method for H(curl)-elliptic problems. We first get the mixed model of H(curl)-elliptic problem by introducing a new intermediate variable.…
Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu…
A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and…
A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the…
This paper is about elliptic and parabolic partial differential operators with discontinuities in the gradient which are compatible with a Finsler norm in a sense to be made precise. Examples of this type of problems arise in a number of…
We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension…
In this work we present an a posteriori analysis for classes of inconsistent, nonconforming schemes approximating elliptic problems. We show the estimates coincide with existing ones for interior penalty type discontinuous Galerkin…
Numerical schemes for Einstein's vacuum equation are developed. Einstein's equation in harmonic gauge is second order symmetric hyperbolic. It is discretized in four-dimensional spacetime by Finite Differences, Finite Elements, and Interior…
A mathematical analysis is presented for a class of interior penalty (IP) discontinuous Galerkin approximations of elliptic boundary value problems. In the framework of the present theory one can derive some overpenalized IP bilinear forms…
Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or…
This paper is devoted to the convergence and optimality analysis of the adaptive Morley element method for the fourth order elliptic problem. A new technique is developed to establish a quasi-orthogonality which is crucial for the…