This thesis presents the numerical algorithms to carry out the face transformation. The main efforts are denoted to the finite volume method (FVM) with the Delaunay triangulation to solve the Laplace equations in the harmonic transformation undergone in face images. The advantages of the FVM with the Delaunay triangulation are: (1) Easy to formulate the linear algebraic equations, (2) Good to retain the geometric and physical properties, (3) less CPU time needed. The numerical and graphical experiments are reported for the face transformations from a female to a male, and vice versa. The computed sequential and absolute errors are and , where N is division number of a pixel into subpixels. Such computed errors coincide with the analysis on the splitting-shooting method (SSM) with piecewise constant interpolation in [Li and Bui, 1998c].

In the dissertation we discuss the rectangular elements, Adini's elements and $p-$order Lagrange elements, which were constructed in the rectangular finite spaces. The special rectangular partitions enable the finite element solutions $u_h$ more efficient in interpolation of the true solution for Elliptic equation $u_I$. The convergence rates of $|u_h-u_I|_1$ are one or two orders higher than the optimal convergence rates. For post-processings we construct higher order interpolation operation $Pi_p$ to reach superconvergence $|u-Pi_p u_h|_1$. To our best knowledge, we at the first time provided the a posteriori interpolant formulas of Adini's elements and biquadratic Lagrange elements to obtain the global superconvergence, and at the first time reported the numerical verification for supercloseness $O(h^4)-O(h^5) $, global superconvergence $O(h^5)$ in $H^1$-norm and the high rates $O(h^6|ln h|)$ in the infinity norm for Poisson's equation(i.e., $-Delta u = f$). Since the finite element methods is fail to deal with the singularity problems, in the dissertation, the combinations of the Ritz-Galerkin method and the finite element methods are used for the singularity problem, i.e., Motz's problem. To couple two methods along their common boundary, we adopt the simplified hybrid, penalty, and penalty plus hybrid techniques. The analysis are made in the dissertation to derive the almost best global superconvergence $O(h^{p+2-delta})$ in $H^1$-norm, $0<delta << 1$, for the combination using $p(geq 2)$-rectangles in the smooth subdomain, and the best global superconvergence $O(h^{3.5})$ in $H^1$-norm for combinations of Adini's elements in the smooth subdomain. The numerical experiments have been carried out for the combinations of the Ritz-Galerkin method and Adini's elements, to verify the theoretical superconvergence derived.