The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

We present new results in the numerical analysis of singularly perturbed convection-diffusion-reaction problems that have appeared in the last five years. Mainly discussing layer-adapted meshes, we present also a survey on stabilization methods, adaptive methods, and on systems of singularly perturbed equations.

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Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

A brief survey on numerical methods for solving singularly perturbed problems

A mixed finite element method for a time-fractional fourth-order partial differential equation

In this article, we propose a mixed finite element method for the two-dimensional Biot's consolidation model of poroelasticity. The new mixed formulation presented herein uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure. This method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger–Reissner formulation for the mechanical subproblem. Optimal a-priori error estimates are proved for both semidiscrete and fully discrete problems when the Raviart–Thomas space for the flow problem and the Arnold–Winther space for the elasticity problem are used. In particular, optimality in the stress, displacement, and pressure has been proved in when the constrained-specific storage coefficient is strictly positive and in the weaker norm when is nonnegative. We also present some of our numerical results.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1189–1210, 2014

Convergence analysis of a new mixed finite element method for Biot's consolidation model

In this paper an anisotropic interpolation theorem is presented that can be easily used to check the anisotropy of an element. A kind of quasi-Wilson element is considered for second-order problems on narrow quadrilateral meshes for which the usual regularity condition ρ K / h K ≥ c 0 > 0 is not satisfied, where h K is the diameter of the element K and ρ K is the radius of the largest inscribed circle in K. Anisotropic error estimates of the interpolation error and the consistency error in the energy norm and the L 2 -norm are given. Furthermore, we give a Poincare inequality on a trapezoid which improves a result of Zenisek.

Anisotropic interpolation and quasi‐Wilson element for narrow quadrilateral meshes

We propose a simple nonconforming virtual element for plate bending problems, which has few local degrees of freedom and provides the optimal convergence in $$H^2$$
 -norm. Moreover, we prove the optimal error estimates in $$H^1$$
 - and $$L^2$$
 -norm. The nonconforming virtual element is constructed for any order of accuracy, but not $$C^0$$
 -continuous. It is worth mentioning that, for the lowest-order case on triangular meshes the simplified nonconforming virtual element coincides with the well-known Morley element, so it can be taken as the extension of the Morley element to polygonal meshes. Finally, we verify the optimal convergence in $$H^2$$
 -norm for the nonconforming virtual element by some numerical tests.

The Morley-Type Virtual Element for Plate Bending Problems

We present a new family of rectangular mixed finite elements for the stress-displacement system of the plane elasticity problem. Based on the theory of mixed finite element methods, we prove that they are stable and obtain error estimates for both the stress field and the displacement field. Using the finite element spaces in this family, an exact sequence is established as a discrete version of the elasticity complex in two dimensions. And the relationship between this discrete version and the original one is shown in a commuting diagram.

Conforming Rectangular Mixed Finite Elements for Elasticity

In this paper, we consider the nonconforming virtual element method (VEM) for the approximation of the time fractional reaction–subdiffusion equation involving the Caputo fractional derivative. For the numerical discrete method of the Caputo fractional derivative, we permit the use of nonuniform time steps, since they are helpful to deal with the non-smooth system. Meanwhile, the nonconforming VEM, which is constructed for any order of accuracy and for very general shaped polygonal and polyhedral meshes, is adopted for the discretization of the spatial direction. By introducing a new Ritz projection operator and using two extended ty$$L^2$$-normpes of continuous and discrete fractional Gronwall inequalities, the optimal error estimates for the spatial semi-discrete and temporal-spatial fully discrete systems are proved detailedly. Besides, the fully discrete scheme is proved to be unconditionally stable with regard to the $$L^2$$- and $$H^1$$-norms, respectively. Finally, some numerical calculations are implemented to verify the theoretical results.

Shaochun Chen

Papers

Anisotropic interpolation and quasi‐Wilson element for narrow quadrilateral meshes

The Morley-Type Virtual Element for Plate Bending Problems

Conforming Rectangular Mixed Finite Elements for Elasticity

Nonconforming Virtual Element Method for the Time Fractional Reaction–Subdiffusion Equation with Non-smooth Data

Anisotropic interpolations with application to nonconforming elements