Well posedness for the Poisson problem on closed Lipschitz manifolds

Michaël Ndjinga () and Marcial Nguemfouo ()
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Michaël Ndjinga: Université Paris-Saclay
Marcial Nguemfouo: University of Yaounde 1

Partial Differential Equations and Applications, 2023, vol. 4, issue 5, 1-19

Abstract: Abstract We study the weak formulation of the Poisson problem on closed Lipschitz manifolds. Lipschitz manifolds do not admit tangent spaces everywhere and the definition of the Laplace–Beltrami operator is more technical than on classical differentiable manifolds (see, e.g., Gesztesy in J Math Sci 172:279–346, 2011). They however arise naturally after the triangulation of a smooth surface for computer vision or simulation purposes. We derive Stokes’ and Green’s theorems as well as a Poincaré’s inequality on Lipschitz manifolds. The existence and uniqueness of weak solutions of the Poisson problem are given in this new framework for both the continuous and discrete problems. As an example of application, numerical results are given for the Poisson problem on the boundary of the unit cube.

Keywords: Lipschitz manifold; Laplace–Beltrami operator; Finite element method; Elliptic equation; Closed surfaces; 58J05; 35J05; 65N30 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00263-x

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