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Meshfree finite difference approximations for functions of the eigenvalues of the Hessian

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Abstract

We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which allows for very complicated domains and a non-uniform distribution of discretisation points. The schemes are monotone, which ensures that they converge to the viscosity solution of the underlying PDE as long as the equation has a comparison principle. Numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds, complex domains, degenerate equations, and singular solutions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ, 07102, USA

    Brittany D. Froese

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Correspondence to Brittany D. Froese.

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This work was partially supported by NSF DMS-1619807.

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Froese, B.D. Meshfree finite difference approximations for functions of the eigenvalues of the Hessian. Numer. Math. 138, 75–99 (2018). https://doi.org/10.1007/s00211-017-0898-2

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