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A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation

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Abstract

In this paper, we study finite element approximations of the viscosity solution of the fully nonlinear Monge-Ampère equation, det(D 2 u) = f (> 0) using the well-known nonconforming Morley element. Our approach is based on the vanishing moment method, which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and Feng (J Sci Comput 38(1):74–98, 2009). The vanishing moment method approximates the Monge-Ampère equation by the fourth order quasilinear equation \({-\epsilon\Delta^2u^\epsilon + {\rm det}(D^2u^\epsilon) = f}\) with appropriate boundary conditions. We develop a finite element scheme using the n-dimensional Morley element introduced in Wang and Xu (Numer Math 103:155–169, 2006) to approximate the regularized fourth order problem in two and three dimensions, and then derive optimal order error estimates.

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Correspondence to Michael Neilan.

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Neilan, M. A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation. Numer. Math. 115, 371–394 (2010). https://doi.org/10.1007/s00211-009-0283-x

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Mathematics Subject Classification (2000)

  • 65N30
  • 65N12
  • 35J60