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Mathematics > Numerical Analysis

arXiv:2301.06805 (math)
[Submitted on 17 Jan 2023]

Title:Stability and guaranteed error control of approximations to the Monge--Ampère equation

Authors:Dietmar Gallistl, Ngoc Tien Tran
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Abstract:This paper analyzes a regularization scheme of the Monge--Ampère equation by uniformly elliptic Hamilton--Jacobi--Bellman equations. The main tools are stability estimates in the $L^\infty$ norm from the theory of viscosity solutions which are independent of the regularization parameter $\varepsilon$. They allow for the uniform convergence of the solution $u_\varepsilon$ to the regularized problem towards the Alexandrov solution $u$ to the Monge--Ampère equation for any nonnegative $L^n$ right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the $L^\infty$ norm for continuously differentiable finite element approximations of $u$ or $u_\varepsilon$.
Subjects: Numerical Analysis (math.NA)
Cite as: arXiv:2301.06805 [math.NA]
  (or arXiv:2301.06805v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2301.06805
Journal reference: Numer. Math. 156 (2024), 107-131
Related DOI: https://doi.org/10.1007/s00211-023-01385-5

Submission history

From: Dietmar Gallistl [view email]
[v1] Tue, 17 Jan 2023 11:03:15 UTC (289 KB)
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