Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. This dissertation investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid methods to reduce accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. Parabolic initial-boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution. For those problems, specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution. This dissertation investigates characteristics of the θ-method and introduces a variable-θ method as a synergistic combination of the Crank-Nicolson (CN) method and the implicit method. It suppresses spurious oscillations, by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness, and maintains as a similar accuracy as the CN method with smooth data. An effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations (the wobble set); the resulting variable-θ method is analyzed for its accuracy and stability. After a theory of morphogenesis in chemical cells was introduced in 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) equations. This dissertation studies a nonoscillatory second-order time-stepping procedure for RD equations incorporating with variable-θ method, as a perturbation of the CN method. We also perform a sensitivity analysis for the numerical solution of RD systems to conclude that it is much more sensitive to the spatial mesh resolution than the temporal one. Moreover, to enhance the spatial approximation of RD equations, this dissertation investigates the averaging scheme, that is, an interpolation of the standard and skewed discrete Laplacian operator and introduce the simple optimizing strategy to minimize the leading truncation error of the scheme.

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