A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under quadrilateral meshes. It has been proven that the recovered gradient converges at a rate $O(h^{1+\rho})$ for $\rho = min(\alpha, 1)$, when the mesh is distorted $O(h^{1+\alpha})$ ($\alpha > 0$) from a regular one. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact.

We try to give a brief survey about using multiple level set methods for identifying piecewise constant or piecewise smooth functions. A general framework is presented. Application using this general framework for different practical problems are shown. We try to show some details in applying the general approach for applications to: image segmentation, optimal shape design, elliptic inverse coefficient identification, electrical impedance tomography and positron emission tomography. Numerical experiments are also presented for some of the problems.

Least-squares principles use artificial "energy" functionals to provide a Rayleigh-Ritz-like setting for the finite element method. These functionals are defined in terms of PDE’s residuals and are not unique. We show that viable methods result from reconciliation of a mathematical setting dictated by the norm-equivalence of least-squares functionals with practicality constraints dictated by the algorithmic design. We identify four universal patterns that arise in this process and develop this paradigm for first-order ADN elliptic systems. Special attention is paid to the effects that each discretization pattern has on the computational and analytic properties of finite element methods, including error estimates, conditioning of the algebraic systems and the existence of efficient preconditioners.

In this paper, we study the Weissman-Taylor rectangular element for the Reissner-Mindlin plate [12] model and provide a convergence analysis for the transverse displacement and the rotation. We show that the element is stable and locking free, thereby improve the results of [8] and [9].

In this paper we systematically derive, via the theory of homogenization, the macroscopic equations for the mechanical behavior of a deformable porous medium saturated with a Newtonian fluid. The derivation is first based on the equations of linear elasticity in the solid, the Stokes equations for the fluid, and suitable conditions at the fluid-solid interface. A detailed comparison between the equations derived here and those by Biot is given. The homogenization approach determines the form of the macroscopic constitutive relationships between variables and shows how to compute the coefficients in these relationships. The derivation is then extended to the nonlinear Navier-Stokes equations for the fluid in the deformable porous medium for the first time. A generalized Forchheimer law is obtained to take into account the nonlinear inertial effects on the flow of the Newtonian fluid through such a medium. Both quasi-static and transient flows are considered in this paper. The properties of the macroscopic coefficients are studied. The computational results show that the macroscopic equations predict well the behavior of the microscopic equations in certain reasonable test cases.

The Reissner-Mindlin model is frequently used by engineers for plates and shells of small to moderate thickness. This model is well known for its "locking" phenomenon so that many numerical approximations behave poorly when the thickness parameter tends to zero. Following the formulation derived by Brezzi and Fortin, we construct a new finite element scheme for the Reissner-Mindlin model using $L^2$ projections onto appropriately-chosen finite element spaces. A superconvergence result is established for the new finite element solutions by using the $L^2$ projections. The superconvergence is based on some regularity assumption for the Reissner-Mindlin model and is applicable to any stable finite element methods with regular but non-uniform finite element partitions.