Real and Complex Analysis

Finite element methods for Maxwell's equations.

A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann–Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine–Gordon equation in light–cone coordinates, the nonlinear Schrodinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrodinger equation is derived.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1997.0077

A unified transform method for solving linear and certain nonlinear PDEs

We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.

/pdf/finite-element-approximation-of-eigenvalue-problems-g9o6nzbplg.pdf

Finite element approximation of eigenvalue problems

A discrete technique of the Schwarz alternating method is presented in this last chapter, to combine the Ritz-Galerkin and finite element methods. This technique is well suited for solving singularity problems in parallel, and requires a little more computation for large overlap of subdomains. The convergence rate of the iterative procedure, which depends upon overlap of subdomains, will be studied. Also a balance strategy will be proposed to couple the iteration number with the element size used in the FEM. For the crack-infinity problem of singularity the total CPU time by the technique in this chapter is much less than that by the nonconforming combination in Chapter 12.

Schwarz Alternating Method

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.

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

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.

https://iopscience.iop.org/article/10.1088/0951-7715/18/4/019/pdf

The nonlinear Schrödinger equation on the half-line

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.

https://www.ams.org/mcom/1992-59-200/S0025-5718-1992-1140646-2/S0025-5718-1992-1140646-2.pdf

Linear finite element methods for planar linear elasticity

We consider a model Poisson problem in ℝd (d = 2, 3) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges (d = 2) or small faces ...

https://www.worldscientific.com/doi/pdf/10.1142/S0218202518500355

Virtual element methods on meshes with small edges or faces

Abstract We present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions.

Li-Yeng Sung

Papers

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

The nonlinear Schrödinger equation on the half-line

Linear finite element methods for planar linear elasticity

Virtual element methods on meshes with small edges or faces

Some Estimates for Virtual Element Methods