Machine Learning based HP-Adaptive Discontinuous Galerkin Methods for Multiphase Flow in Porous Media
B. Kane · 2024
Summary Modeling and simulation of multiphase flows arising in environmental problems such as infiltration and remediation problems is complex and requires careful numerical treatment due to the strong heterogeneity of the porous media. The discretization of the PDE’s arising from this problem class requires locally conservative methods in order to be able to follow small concentration. Discontinuous Galerkin (DG), Finite Volume and Mixed Finite Element, are examples of discretization methods that achieve local conservation at the element level. DG methods are based on weak formulations with finite dimensional piecewise polynomial solution space and test function space. The main difference with classical Finite Element methods is that the finite element function space corresponding to DG methods consists of piecewise polynomials which are allowed to be completely discontinuous across element interfaces. Numerical fluxes and penalty terms are added in order to enforce weakly the continuity of the solution and the boundary conditions. DG methods present attractive features such as an inherent local and global conservation, a high-order accuracy, a high parallel efficiency and a geometric flexibility (unstructured meshes and non¬conforming grids) allowing an easier local hp adaptivity. The hp-adaptive methods combine both h-adaptive and p-adaptive methods. This strategy allows to refine the mesh when the solution is estimated to be rough (e.g. near discontinuities) and increase the polynomial degree when the solution is estimated to be smooth. This helps to compensate the increased computational cost for complex models. Here, we extend our previous work on the matter by introducing a machine learning based method to drive the p-adaptive process. A clustering method namely the Gaussian mixture method (GMM) is used to split the domain in different regions. The mesh refinement is driven by classical error indicators while the polynomial degree modification is based on the GMM. The implementation is based on Dune-FemPy and the code is provided as a Jupyter notebook accessible through a Docker container.