A variational meshfree method for solving time-discrete diffusion equations

摘要

A meshfree method is developed for solving time-discrete diffusion equations that arise in models in brain research. Important criteria for a suitable method are flexibility with respect to domain geometry and the ability to work with very small moving sources requiring easy refinement possibilities. One part of the work concerns a meshfree discretization of the modified Helmholtz equation based on the related minimization problem and a local least-squares function approximation. In a second part, a node choosing algorithm is presented that moves around randomly distributed nodes for optimizing the node distribution and varying the node density as needed. The method is illustrated by two numerical tests.