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FEM-RBF: A geometrically flexible, efficient numerical solution technique for partial differential equations with mixed regularity

Final Report Summary - FEMRBF (FEM-RBF: A geometrically flexible, efficient numerical solution technique for partial differential equations with mixed regularity)

Problems that exhibit solutions with different regularities on localized regions are often encountered in many scientific applications related to solid mechanics, acoustics, and electromagnetic. In those applications, we often have to deal with solving non-periodic time-independent partial differential equations (PDEs) on complex geometries. Furthermore, due to the underlying physical properties such as defects, cracks, and/or material imperfections, solutions may be hard to resolve around the troubled regions. Methods for numerically simulating solutions of such problems must be able to efficiently resolve features around the defects. In the mean time, solutions away from the defects must also be computed as accurate as possible. The goal of this project is to develop and analyze techniques suitable for solving such problems with optimal computational costs and respectable accuracy by combining two powerful methods: finite element (FE) and radial basis function (RBF).
Global approximation methods such as radial basis function methods have proven extremely useful in the numerical solution of boundary value problems in high dimensions. They can be implemented on a flexible mesh and nodes adaptivity can be easily accomplished by adding or deleting points as necessary. RBF methods have been widely used for scattered data interpolation in high dimensions. Recently RBF collocation methods for PDEs, based on global, non-polynomial interpolants, have been developed. The RBF methods are referred to as meshfree methods since they may be implemented on scattered sets of collocation sites (commonly called centers) and are not tied to structured grids as are pseudospectral methods. In this way, RBF methods overcome some limitations of pseudospectral methods. RBF collocation methods for steady PDEs have become well established. Although the adaptive RBF method is very promising to be used as a main solver for problems with defects, hybridizing it with finite element methods may have big impact on overall computational cost and efficiency. Around the defect regions, where solutions are less smooth and even ”classical” solutions do not exist, the finite element methods will be utilized. Away from those regions, where solutions are smooth and classical solutions can be computed in collocation way, RBF methods will be used. The hybrid method can be constructed by decomposing the computational domains into FEM and RBF sub domains. Those sub domains will result in interface regions, i.e regions where two different methods meet. Solutions must match in terms of continuity and normal derivatives there. This approach combines the strengths of both the finite element methods in the regions, where solutions have less regularities, with the flexibility of RBF methods where required for smooth solution regions. We numerically study and analyze techniques to couple the two methods in a stable way.