OPTIMAL APPROXIMATION SPACES FOR SOLVING PROBLEMS WITH ROUGH COEFFICIENTS

Abstract

The finite element method has been widely used to solve partial

differential equations by both engineers and mathematicians for the

last several decades. This is due to its well-known effectiveness

when applied to a wide variety of problems. However, it has some

practical drawbacks. One of them is the need for meshing. Another is

that it uses polynomials as the approximation basis functions.

Commonly, polynomials are also used by other numerical methods for

partial differential equations, such as the finite difference method

and the spectral method. Nevertheless, polynomial approximations are

not always effective, especially for problems with rough

coefficients. In the dissertation, a suitable approximation space

for the solution of elliptic problems with rough coefficients has

been found, which is named as generalized <italic>L</italic>-spline space. Theoretically, I have developed generalized <italic>L</italic>-spline approximation

spaces, where <italic>L</italic> is an operator of order <italic>m</italic> with rough coefficients, have proved the interpolation error estimate, and have also proved that the generalized <italic>L</italic>-spline space is an optimal approximation space for the problem <italic>L*Lu=f</italic> with certain operator <italic>L</italic>, by using <italic>n</italic>-widths as the criteria. Numerically, two problems have been tested and the relevant error estimate results are consistent with the shown theoretical results.

Meshless methods are newly developed numerical methods for solving

partial differential equations. These methods partially eliminate

the need of meshing. Meshless methods are considered to have great

potential. However, the need for effective quadrature schemes is a

major issue concerning meshless methods. In our recently published

paper, we consider the approximation of the Neumann problem by

meshless methods, and show that the approximation is inaccurate if

nothing special (beyond accuracy) is assumed about the numerical

integration. We then identify a condition - referred to as the zero

row sum condition. This, together with accuracy, ensure the

quadrature error is small. The row sum condition can be achieved by

changing the diagonal elements of the stiffness matrix. Under row

sum condition we derive an energy norm error estimate for the

numerical solution with quadrature. In the dissertation, meshless

methods are discussed and quadrature issue is explained. Two

numerical experiments are presented in details. Both theoretical and

numerical results indicate that the error has two components; one

due to the meshless methods approximation and the other due to

quadrature.