We derive an a posteriori error estimator giving a computable upper bound on the error in the energy norm for finite element approximation using the non-conforming rotated $\mathbb{Q}_1$ finite element. It is shown that the estimator also gives a local lower bound up to a generic constant. The bounds do not require additional assumptions on the regularity of the true solution of the underlying elliptic problem and, the mesh is only required to be locally quasi-uniform and may consist of general, non-affine convex quadrilateral elements.

In [20], we analytically identified natural superconvergent points of function values and gradients for several popular three-dimensional polynomial finite elements via an orthogonal decomposition. This paper focuses on the detailed process for determining the superconvergent points of pentahedral and tetrahedral elements.

A technique of orthogonality correction in an element is introduced and applied to superconvergence analysis in finite element method. Ultraconvergence results for rectangular elements of odd degree $n \geq 3$ are derived in the case of variable coefficients.

We give an overview of superconvergence phenomena in the finite element method for solving three-dimensional problems, in particular, for elliptic boundary value problems of second order over uniform meshes. Some difficulties with superconvergence on tetrahedral meshes are presented as well. For a given positive integer $m$ we prove that there is no tetrahedralization of $R^3$ whose all edges are $m$-valent.

In this paper convergence on equidistributing meshes is investigated. Equidistributing meshes, or more generally approximate equidistributing meshes, are constructed through the well-known equidistribution principle and a so-called adaptation (or monitor) function which is defined based on estimates on interpolation error for polynomial preserving operators. Detailed convergence analysis is given for finite element solution of singularly perturbed two-point boundary value problems without turning points. Illustrative numerical results are given for a convection-diffusion problem and a reaction-diffusion problem.

This paper presents a high order local discontinuous Galerkin time-domain method for solving time dependent Schrödinger equations. After rewriting the Schrödinger equation in terms of a first order system of equations, a numerical flux is constructed to preserve the conservative property for the density of the particle described. Numerical results for a model square potential scattering problem is included to demonstrate the high order accuracy of the proposed numerical method.

We consider the discretization in time of a parabolic equation, using a representation of the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. This reduces the problem to a finite set of elliptic equations, which may be solved in parallel. The procedure is combined with finite element discretization in the spatial variables. The method is also applied to some parabolic type evolution equations with memory.

Convergence and superconvergence of the interpolated coefficient finite element method (ICFEM) are discussed as the ICFEM reduces the computation cost greatly. Further, the ICFEM is implemented to compute the multiple solutions of some semilinear elliptic problems.

In this paper, we consider a two-dimensional parabolic equation with two small parameters. These small parameters make the underlying problem containing multiple scales over the whole problem domain. By using the maximum principle with carefully chosen barrier functions, we obtain the pointwise derivative estimates of arbitrary order, from which an anisotropic mesh is constructed. This mesh uses very finer mesh inside the small scale regions (where the boundary layers are located) than elsewhere (large scale regions). A fully discrete backward difference Galerkin scheme based on this mesh with arbitrary $k$-th ($k \geq 1$) order conforming rectangular elements is discussed. Note that the standard finite element analysis technique can not be used directly for such highly nonuniform anisotropic meshes because of the violation of the quasi-uniformity assumption. Then we use the integral identity superconvergence technique to prove the optimal uniform convergence $O(N^{-(k+1)} + M^{-1})$ in the discrete $L^2$-norm, where $N$ and $M$ are the number of partitions in the spatial (same in both the $x$- and $y$-directions) and time directions, respectively.