Finite Volume Scheme for a Transmission Elliptic Problem with the Fictitious Domain Method Principle

Abstract

This thesis addresses elliptic transmission problems occurring in domains composed of different materials, which lead to discontinuous coefficients in the partial differential equations. To handle complex geometries and interface conditions, the fictitious domain method is employed, allowing the use of a regular mesh regardless of domain irregularities. The finite volume method is used to enforce local conservation of fluxes. The work includes a rigorous mathematical analysis based on Sobolev spaces, a variational formulation, and the implementation of a stable and accurate numerical scheme. Numerical tests demonstrate the convergence, accuracy, and stability of the method.