Michael Neilan

Conforming and divergence-free Stokes elements on general triangular meshes

We present a family of conforming finite elements for the Stokes problem on general triangular meshes in two dimensions. The lowest order case consists of enriched piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure. We show that the elements satisfy the inf-sup condition and converges with order k for both the velocity and pressure. Moreover, the pressure space is exactly the divergence of the corresponding space for the velocity. Therefore the discretely divergence-free functions are divergence-free pointwise. We also show how the proposed elements are related to a class of C1 elements through the use of a discrete de Rham complex.

{C}^0

In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.

penalty methods for the fully nonlinear Monge-Ampère equation

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampere equation

Mixed Finite Element Methods for the Fully Nonlinear Monge-Ampère Equation Based on the Vanishing Moment Method

\det(D^2u^0)=f\,(>0)

Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations

based on the vanishing moment method which was proposed recently by the authors in [X. Feng and M. Neilan, J. Scient. Comp., DOI 10.1007/s10915-008-9221-9, 2008]. In this approach, the second-order fully nonlinear Monge-Ampere equation is approximated by the fourth order quasilinear equation

Stokes Complexes and the Construction of Stable Finite Elements with Pointwise Mass Conservation

-\varepsilon\Delta^2 u^\varepsilon + \det{D^2u^\varepsilon}=f

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

. It was proved in [X. Feng, Trans. AMS, submitted] that the solution

A

u^\varepsilon

\mathcal{C}^0

converges to the unique convex viscosity solution

Interior Penalty Method for a Fourth Order Elliptic Singular Perturbation Problem

u^0