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Convergence of adaptive discontinuous Galerkin and C⁰-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations

Kawecki, EL; Smears, I; (2021) Convergence of adaptive discontinuous Galerkin and C⁰-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations. Foundations of Computational Mathematics 10.1007/s10208-021-09493-0. (In press). Green open access

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Abstract

We prove the convergence of adaptive discontinuous Galerkin and C0-interior penalty methods for fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We consider a broad family of methods on adaptively refined conforming simplicial meshes in two and three space dimensions, with fixed but arbitrary polynomial degrees greater than or equal to two. A key ingredient of our approach is a novel intrinsic characterization of the limit space that enables us to identify the weak limits of bounded sequences of nonconforming finite element functions. We provide a detailed theory for the limit space, and also some original auxiliary functions spaces, that is of independent interest to adaptive nonconforming methods for more general problems, including Poincaré and trace inequalities, a proof of the density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, approximation results by finite element functions and weak convergence results.

Type: Article
Title: Convergence of adaptive discontinuous Galerkin and C⁰-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s10208-021-09493-0
Publisher version: https://doi.org/10.1007/s10208-021-09493-0
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10119218
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