Li-Yeng Sung

C0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains

C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are analyzed in this paper. A post-processing procedure that can generate C1 approximate solutions from the C0 approximate solutions is presented. New C0 interior penalty methods based on the techniques involved in the post-processing procedure are introduced. These new methods are applicable to rough right-hand sides.

Linear finite element methods for planar linear elasticity

A linear nonconforming (conforming) displacement finite element method for the pure displacement (pure traction) problem in two-dimensional linear elasticity for a homogeneous isotropic elastic material is considered. In the case of a convex polygonal configuration domain, error estimates in the energy (L[sup 2]) norm are obtained. The convergence rate does not deteriorate for nearly incompressible material. Furthermore, the convergence analysis does not rely on the theory of saddle point problems. 22 refs.

The nonlinear Schrödinger equation on the half-line

Assuming that the solution q(x, t) of the nonlinear Schrodinger equation on the half-line exists, it has been shown in Fokas (2002 Commun. Math. Phys. 230 1–39) that q(x, t) can be represented in terms of the solution of a matrix Riemann–Hilbert (RH) problem formulated in the complex k-plane. The jump matrix of this RH problem has explicit x, t dependence and it is defined in terms of the scalar functions {a(k), b(k), A(k), B(k)} referred to as spectral functions. The functions a(k) and b(k) are defined in terms of q0(x) = q(x,0), while the functions A(k) and B(k) are defined in terms of g0(t) = q(0,t) and g1(t) = qx(0,t). The spectral functions are not independent but they satisfy an algebraic global relation. Here we first prove that if there exist spectral functions satisfying this global relation, then the function q(x, t) defined in terms of the above RH problem exists globally and solves the nonlinear Schrodinger equation, and furthermore q(x, 0) = q0(x), q(0, t) = g0(t) and qx(0, t) = g1(t). We then show that, given appropriate initial and boundary conditions, it is possible to construct such spectral functions through the solution of a nonlinear Volterra integral equation whose solution exists globally. We also show that for a particular class of boundary conditions it is possible to bypass this nonlinear equation and to compute the spectral functions using only the algebraic manipulation of the global relation; thus for this particular class of boundary conditions, which we call linearizable, the problem on the half-line can be solved as effectively as the problem on the line. An example of a linearizable boundary condition is qx(0, t) − ρq(0, t) = 0 where ρ is a real constant.

{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

A new numerical method for computing the divergence-free part of the solution of the time-harmonic Maxwell equations is studied in this paper. It is based on a discretization that uses the locally divergence-free Crouzeix-Raviart nonconforming P 1 vector fields and includes a consistency term involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive e) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

A locally divergence-free nonconforming finite element method for the time-harmonic maxwell equations

An interior penalty method for certain two-dimensional curl-curl problems is investigated in this paper. This method computes the divergence-free part of the solution using locally divergence-free discontinuous

A Locally Divergence-Free Interior Penalty Method for Two-Dimensional Curl-Curl Problems

P_1

Multigrid Algorithms for C 0 Interior Penalty Methods

vector fields on graded meshes. It has optimal order convergence (up to an arbitrarily small

A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem

\epsilon

Balancing Domain Decomposition for Nonconforming Plate Elements

) for the source problem and the eigenproblem. Results of numerical experiments that corroborate the theoretical results are also presented.