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This paper surveys recent numerical advances in the phase field method forgeometric surface evolution and related geometric nonlinear partialdifferential equations (PDEs). Instead of describing technical details ofvarious numerical methods and their analyses, the paper presents a holisticoverview about the main ideas of phase field modeling, its mathematicalfoundation, and relationships between the phase field formalism and othermathematical formalisms for geometric moving interface problems, as well as thecurrent state-of-the-art of numerical approximations of various phase fieldmodels with an emphasis on discussing the main ideas of numerical analysistechniques. The paper also reviews recent development on adaptive grid methodsand various applications of the phase field modeling and their numericalmethods in materials science, fluid mechanics, biology and image science.

The Phase Field Method for Geometric Moving Interfaces and Their Numerical Approximations