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Multigrid methods for convergent mixed finite difference scheme for Monge–Ampère equation

  • Special Issue IMG 2016
  • Published:
Computing and Visualization in Science

Abstract

We propose multigrid methods for convergent mixed finite difference discretization for the two dimensional Monge–Ampère equation. We apply mixed standard 7-point stencil and semi-Lagrangian wide stencil discretization, such that the numerical solution is guaranteed to converge to the viscosity solution of the Monge–Ampère equation. We investigate multigrid methods for two scenarios. The first scenario considers applying standard 7-point stencil discretization on the entire computational domain. We use full approximation scheme with four-directional alternating line smoothers. The second scenario considers the more general mixed stencil discretization and is used for the linearized problem. We propose a coarsening strategy where wide stencil points are set as coarse grid points. Linear interpolation is applied on the entire computational domain. At wide stencil points, injection as the restriction yields a good coarse grid correction. Numerical experiments show that the convergence rates of the proposed multigrid methods are mesh-independent.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

    Yangang Chen

  2. David R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

    Justin W. L. Wan

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  1. Yangang Chen
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  2. Justin W. L. Wan
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Correspondence to Yangang Chen.

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Communicated by Gabriel Wittum.

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Chen, Y., Wan, J.W.L. Multigrid methods for convergent mixed finite difference scheme for Monge–Ampère equation. Comput. Visual Sci. 22, 27–41 (2019). https://doi.org/10.1007/s00791-017-0284-8

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