SURFACE FINITE ELEMENT METHOD

How finite elements capture the geometry of a domain

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SURFACE PDES

• PDE modelling of natural phenomena on surfaces

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SURFACE DIFFERENTIAL OPERATORS

• Differential operators can be generalized on surfaces

• E.g. Surface Poisson equation :

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WEAK FORMULATION & SURFACE FEM

• Weak formulation of :

• Surface finite element method :

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EXAMPLE I : POISSON EQUATION

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EXAMPLE 2 : EIGENFUNCTIONS OF LAPLACIAN

EXAMPLE 2 : EIGENFUNCTIONS OF LAPLACIAN

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EXAMPLE 2 : EIGENFUNCTIONS OF LAPLACIAN

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EXAMPLE 2 : EIGENFUNCTIONS OF LAPLACIAN

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REFERENCE

• Surface finite element method

1. G.Dziuk, C.M.Elliott, Finite element methods for surfaces PDEs, ActaNumerica (2013) pp. 289-396

• Laplacian eigenfunctions on surfaces

1. M.Reuter, F.Wolter, N.Peinecke, Laplace-Beltrami spectra as ‘Shape-DNA’ of surfaces and solids, Computer-Aided Design 38 (2006), pp. 342-366

2. Y.Canzani, Analysis on Manifolds via the Laplacian, available at : http://www.math.mcgill.ca/toth/spectral%20geometry.pdf

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CONCLUSION

• Natural extension of FEM• Difficult in :

1. Mesh refinement2. High order approximation