Interior Penalty Discontinuous Galerkin Based Isogeometric Analysis for Allen-Cahn Equations on Surfaces (Communications in Computational Physics, 2020)
We propose a method that combines Isogeometric Analysis (IGA) with theinterior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahnequation, arising from phase transition in materials science, on three-dimensional (3D)surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., weemploy the standard IGA within each patch, and employ the IPDG method across thepatch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs)on (3D) surfaces and the IPDG method is used to glue the multiple patches together toget the right solution. Our method takes advantage of both IGA and the IPDG method,which allows us to design a superior semi-discrete (in time) IPDG scheme. First andmost importantly, the time-consuming mesh generation process in traditional Finite ElementAnalysis (FEA) is no longer necessary and refinements, including h-refinementand p-refinement which both maintain the original geometry, can be easily performedat any level. Moreover, the flexibility of the IPDG method makes our method veryeasy to handle cases with non-conforming patches and different degrees across thepatch interfaces. Additionally, the geometrical error is eliminated (for all conic sections)or significantly reduced at the beginning due to the geometric flexibility of IGAbasis functions, especially the use of multiple patches. Finally, this method can beeasily formulated and implemented. We present our semi-discrete IPDG scheme aftergenerally describe the problem, and then briefly introduce the time marching methodemployed in this paper. Theoretical analysis is carried out to show that our methodsatisfies a discrete energy law, and achieves the optimal convergence rate with respectto the $L^2$ norm. Furthermore, we propose an elliptic projection operator on (3D) surfacesand prove an approximation error estimate which are vital for us to obtain theerror estimate in the $L^2$ norm. Numerical tests are given to validate the theory andgauge the good performance of our method.