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Pointwise rates of convergence for the Oliker–Prussner method for the Monge–Ampère equation

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Abstract

We study the Oliker–Prussner method exploiting its geometric nature. We derive discrete stability and continuous dependence estimates in the max-norm by using a discrete Alexandroff estimate and the Brunn–Minkowski inequality. We show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain; nodes still conform to the domain boundary. . This gives a suitable notion of operator consistency which, combined with stability, leads to pointwise rates of convergence for classical and non-classical solutions of the Monge–Ampère equation.

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Authors and Affiliations

  1. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, USA

    Ricardo H. Nochetto

  2. Department of Mathematics, Rutgers University, New Brunswick, USA

    Wujun Zhang

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  1. Ricardo H. Nochetto
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  2. Wujun Zhang
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Correspondence to Wujun Zhang.

Additional information

Both authors were partially supported by NSF Grants DMS-1109325 and DMS-1411808. The second author was also partially supported by the Brin Postdoctoral Fellowship of the University of Maryland and the start up fund of Rutgers University.

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Nochetto, R.H., Zhang, W. Pointwise rates of convergence for the Oliker–Prussner method for the Monge–Ampère equation. Numer. Math. 141, 253–288 (2019). https://doi.org/10.1007/s00211-018-0988-9

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  • DOI: https://doi.org/10.1007/s00211-018-0988-9

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