Diffuse Domain Methods with Dirichlet Boundary Conditions

Abstract: The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates the problem on a larger, simple domain where the complex geometry is represented by a smooth phase-field function. This paper introduces and analyses several new DDM methods for solving problems with Dirichlet boundary conditions. We derive two new methods from the mixed formulation of the governing equations. This approach transforms the essential Dirichlet conditions into natural boundary conditions. Additionally, we develop coercive formulations based on Nitsche's method, and provide proofs of coercivity for all new and key existing approximations. Numerical experiments demonstrate the improved accuracy of the new methods, and reveal the balance between $L^2$ and $H^1$ errors. The practical effectiveness of this approach is demonstrated through the simulation of the incompressible Navier-Stokes equations on a benchmark fluid dynamics problems.