Porous medium equation (PME) has been found in many applications of the physical sciences. The equation is nonlinear, degenerate, and in many situations has a free boundary, which altogether pose great challenges for mathematical and numerical analyses. In contrast with the mathematical development of PME, which began in the 1950s and has since had much success, studies of numerical solution did not appear until the 1980s. Though a significant progress has been made since then for the 1D setting, only limited success has been observed for 2D cases. In this dissertation, we will propose several moving mesh methods which improve the accuracy and convergence order of the PME numerical solution.
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