A numerical scheme for free boundary problems that are gradient flows for the area functional

Uwe F. Mayer

Abstract: Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow to treat such free boundary problems in both R² and R³. The advantage of such an approach is the re-usability of much of the setup for all of the different problems. As an example of the method we will compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity.

1991 Subject Classification: 35R35, 65C20, 65R20, 82C26

Key words and phrases: Numerical Schemes, Free Boundary Problem, Motion by Mean Curvature, Gradient flow, Mullins-Sekerka flow, Hele-Shaw flow

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