UC Berkeley

In this thesis we prove maximum principles for high order accurate, lattice-based finite difference discretizations of the Laplace operator in one, two, and three dimensions. The primary finding of this work is that with the introduction of carefully constructed perturbations, we can recover "local" maximum principles for high order accurate operators which are similar to those that naturally hold for second order accurate operators. From these local maximum principles we prove global results which are discrete analogues of the classical weak maximum principle.We first discuss precisely why the typical discrete maximum principle fails for high order accurate discrete second derivative operators in one dimension. We then introduce a quantity which is the maximum of several perturbations of a grid function, and we establish a local maximum principle using these perturbations. We show that the proof can be reduced to the satisfaction of a system of linear inequalities. Next, we show how this proof can be interpreted as a "smoothing" operation on certain basis functions.To extend this machinery to higher dimensions, we obtain two generalizations of the one dimensional results. The first is showing the equivalence of the perturbed variable local maximum principle and a certain linear program. The second is a higher dimensional version of the basis function smoothing argument. We combine these results into an inductive procedure which simultaneously builds up perturbations and increasingly accurate Laplacian stencils which satisfy an associated local maximum principle. Using this procedure, we generate sequences of high order accurate Laplacians in two and three dimensions.Lastly, we obtain global maximum principles from these local results, and prove stability estimates using comparison functions. We conclude with a discussion of future research.