Li, Qiaoluan HelenThe 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.DissertationMathematics