FEM-RBF: A Geometrically Flexible, Efficient Numerical Solution Technique for Partial Differential Equations with Mixed Regularity
ثبت نشده
چکیده
1 About This Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Hybrid FEM-RBF scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.1 One Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Two Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 RBF Interpolant and Differentiation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5 PDE Collocation in the RBF Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.1 Finite Difference Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.1 RBF-FD as a Full Poisson Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2 Notes on Cases when nloc = N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.3 RBF-PU as a Full Poisson Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7 Hybrid FEM-RBF for Smooth Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7.1 Numerical Experiments in 1 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.2 Numerical Experiments in 2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 8 On-going Software Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 10 MATLAB Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 11 MATLAB Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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