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arXiv · 1611.02786

Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Amp\`{e}re equation

Ricardo H. Nochetto·Wujun Zhang
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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-Amp\`{e}re equation.

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Ricardo H. Nochetto, Wujun Zhang. 2016-11-09. Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Amp\`{e}re equation. https://doi.org/10.1007/s00211-018-0988-9

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