Volume 55, Number 2, March-April 2021
|Page(s)||449 - 478|
|Published online||15 March 2021|
Unified analysis of discontinuous Galerkin and C0-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
* Corresponding author: email@example.com
Accepted: 24 November 2020
We provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and C0-IP finite element approximations of fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions in H2 of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for the a priori error analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda–Talenti inequality to piecewise polynomial vector fields.
Mathematics Subject Classification: 65N30 / 65N15 / 65N12
Key words: Fully nonlinear PDE / Hamilton–Jacobi–Bellman and Isaacs equations / Cordes condition / nonconforming finite element methods / error analysis
© EDP Sciences, SMAI 2021
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