Numerical solution to the interface problem in a general domain using Moser's deformation method

摘要

When seeking a solution to the interface problem, the potential theory to represent solution has been widely used. As the solution representation of the interface problem is only well-known for domains with simple inclusion, such as a disk, conformal mapping is often used to transform the arbitrary inclusion to a manageable inclusion to utilize the known solution representation. This paper proposes the construction of a conformal mapping based on Moser's grid deformation method that generates a transformation with positive Jacobian determinant. By establishing an optimal control problem that minimizes the dissimilarity between transformed and reference images with desired constraints, we constructed an appropriate conformal mapping. Several numerical results prove the validity of our approach.