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New $C^0$ interior penalty method for Monge-Ampère equations

Tianyang Chu, Hailong Guo, Zhimin Zhang

Abstract

Monge-Ampère equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the $C^0$ interior penalty method to approximation the viscosity solution of the Monge-Ampère equation. The new methods is inspired from the discrete Miranda-Talenti estimate. Based on the vanishing moment representation, we approximate the Monge-Ampère equation by the fourth order semi-linear equation with some additional boundary conditions. We will use the discrete Miranda-Talenti estimates to ensure the well-posedness of the numerical scheme and derive the error estimates.

New $C^0$ interior penalty method for Monge-Ampère equations

Abstract

Monge-Ampère equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the interior penalty method to approximation the viscosity solution of the Monge-Ampère equation. The new methods is inspired from the discrete Miranda-Talenti estimate. Based on the vanishing moment representation, we approximate the Monge-Ampère equation by the fourth order semi-linear equation with some additional boundary conditions. We will use the discrete Miranda-Talenti estimates to ensure the well-posedness of the numerical scheme and derive the error estimates.
Paper Structure (18 sections, 8 theorems, 62 equations, 3 figures, 4 tables)

This paper contains 18 sections, 8 theorems, 62 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Discrete Miranda-Talenti estimate
  5. Linearization and finite element approximation
  6. $C^0$IP methods for the model equation
  7. Numerical scheme
  8. Well-posedness and $H^2$ error estimate for the numerical scheme
  9. Numerical experiments
  10. Two dimensional numerical experiments
  11. Numerical Example I
  12. Numerical example II
  13. Numerical Example III
  14. Three dimensional numerical experiments
  15. Numerical example IV
  16. ...and 3 more sections

Key Result

Theorem 2.1

Suppose $\varOmega \subset \mathbb{R}^d$ is a bounded convex domain. Then, for all $v \in H^2(\varOmega) \cap H_0^1(\varOmega)$, the following inequality holds:

Figures (3)

  • Figure 1: Plot of error with respect to $\epsilon$. The first column is the numerical result for quadratic element and the second column is the numerical result of cubic element.
  • Figure 2: Plot of the computed viscosity solution using quadratic element with $\epsilon=0.005$ and $h = 1/64$ in 2D.
  • Figure 3: Plot of the computed viscosity solution using quadratic element with $\epsilon=0.005$ and $h = 1/32$ in 3D: (a) $x$-slices at $x =0.25, 0.5, 0.75$; (b) $y$-slices at $y =0.25, 0.5, 0.75$.

Theorems & Definitions (15)

  • Theorem 2.1: Miranda-Talenti estimategris1985maps2000
  • Theorem 2.2: Discrete Miranda-Talenti estimates
  • Theorem 2.3: Discrete Sobolev inequality
  • Proof 1
  • Theorem 3.1
  • Proof 2
  • Remark 3.2
  • Theorem 3.3
  • Proof 3
  • Lemma 4.1
  • ...and 5 more