In this article, we demonstrate how one can improve the numerical solutions of singularly perturbed problems involving multiple boundary layers by using a combination of analytic and numerical tools. Incorporating the structures of boundary layers into finite element spaces can improve the accuracy of approximate solutions and result in significant simplifications. We discuss here convection-diffusion equations in the case where both ordinary and parabolic boundary layers are present.

A basis for the quadratic ($P_2$) nonconforming element of Fortin and Soulie on triangles is introduced. The local and global interpolation operators are defined. Error estimates of optimal order are derived in both broken energy and $L^2(\Omega)$-norms for second-order elliptic problems. Brief numerical results are also shown.

In this work we discuss variants of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead use piecewise constant level set functions, and let interfaces be represented by discontinuities. Some of the properties of the standard level set function are preserved in the proposed method. Using the methods for interface problems, we need to minimize a smooth convex functional under a constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth and locally convex. We show numerical results using the methods for segmentation of digital images.

We investigate the Korn first inequality for quadrilateral nonconforming finite elements of first order approximation properties and clarify the dependence of the constant in this inequality on the discretization parameter $h$. Then we use the nonconforming elements for approximating the velocity in a discretization of the Stokes equations with boundary conditions involving surface forces and, using the result on the Korn inequality, we prove error estimates which are optimal for the pressure and suboptimal for the velocity.

The aim of this paper is to present a general algorithm for the branching solution of nonlinear operator equations in a Hilbert space, namely the $k$-order Taylor expansion algorithm, $k \geq 1$. The standard Galerkin method can be viewed as the 1-order Taylor expansion algorithm; while the optimum nonlinear Galerkin method can be viewed as the 2-order Taylor expansion algorithm. The general algorithm is then applied to the study of the numerical approximations for the steady Navier-Stokes equations. Finally, the theoretical analysis and numerical experiments show that, in some situations, the optimum nonlinear Galerkin method provides higher convergence rate than the standard Galerkin method and the nonlinear Galerkin method.

In this paper we study the numerical solution for an $p$-Laplacian type of evolution system $H_t + \nabla \times [|\nabla \times H|^{p-2} \nabla \times H] = F (x, t)$, $p > 2$ in two space dimensions. For large $p$ this system is an approximation of Bean's critical-state model for type-II superconductors. By introducing suitable transformation, the system is equivalent to a nonlinear parabolic equation. For the nonlinear parabolic problem we obtain the numerical solution by combining approximation schemes for the linear equation and the nonlinear semigroup. The convergence and stability of the scheme are proved. Finally, a numerical experiment is presented.