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Convergence of adaptive discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations

Ellya L. Kawecki, Iain Smears

TL;DR

The paper addresses convergence of adaptive broad-family DG and $C^0$-IP methods for fully nonlinear second-order HJB and Isaacs equations under Cordes coefficients in two and three dimensions. It introduces an intrinsic limit-space framework that characterizes the limit behavior of nonconforming finite element sequences without requiring nested conforming spaces, and it extends the Cordes-renormalized operator $F_{\gamma}$ to a robust variational setting. The authors prove plain convergence of the adaptive schemes and establish reliability/efficiency of a posteriori estimators, along with strong properties of the limit problem, including Hessian symmetry and approximation by quadratic polynomials. The results provide a rigorous foundation for stable, efficient adaptive methods for fully nonlinear PDEs in applications such as stochastic control and differential games, while offering a general framework for nonconforming adaptivity beyond HJB/Isaacs problems.

Abstract

We prove the convergence of adaptive discontinuous Galerkin and $C^0$-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 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.

Convergence of adaptive discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations

TL;DR

The paper addresses convergence of adaptive broad-family DG and -IP methods for fully nonlinear second-order HJB and Isaacs equations under Cordes coefficients in two and three dimensions. It introduces an intrinsic limit-space framework that characterizes the limit behavior of nonconforming finite element sequences without requiring nested conforming spaces, and it extends the Cordes-renormalized operator to a robust variational setting. The authors prove plain convergence of the adaptive schemes and establish reliability/efficiency of a posteriori estimators, along with strong properties of the limit problem, including Hessian symmetry and approximation by quadratic polynomials. The results provide a rigorous foundation for stable, efficient adaptive methods for fully nonlinear PDEs in applications such as stochastic control and differential games, while offering a general framework for nonconforming adaptivity beyond HJB/Isaacs problems.

Abstract

We prove the convergence of adaptive discontinuous Galerkin and -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 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.

Paper Structure

This paper contains 27 sections, 26 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Derivatives and traces of functions of bounded variation.
  4. Jump and average operators.
  5. Tangential derivatives.
  6. Finite element spaces.
  7. Norms.
  8. Lifting operators.
  9. Variational formulation of the problem and adaptive finite element approximation
  10. Variational formulation of the problem
  11. Numerical discretizations and error estimators
  12. Analysis of stabilization terms.
  13. Reliable and efficient a posteriori error estimator.
  14. Adaptive algorithm and main result
  15. Main result.
  16. ...and 12 more sections

Key Result

Lemma 3.1

A function $v\in H^2(\Omega)$ satisfies $F[v]\leq 0$ pointwise a.e. in $\Omega$ if and only if $F_{\gamma}[v]\leq 0$ pointwise a.e. in $\Omega$. Furthermore, a function $v\in H^2(\Omega)$ satisfies $F[v]\geq 0$ pointwise a.e. in $\Omega$ if and only if $F_{\gamma}[v]\geq 0$ pointwise a.e. in $\Omega

Figures (1)

  • Figure 1: Degrees of freedom of the cubic Hsieh--Clough--Tocher (HCT) macro-element in two (left) and three (right) space dimensions on a reference element. The basis functions are $C^1$-regular and piecewise cubic with respect to subdivisions of the element into subsimplices CloughTocher66DouglasDupontPercellScott79WorseyFarin87. Solid dots represent degrees of freedom associated to point values, the circles represent gradient values, and the arrows represent directional derivative values.

Theorems & Definitions (57)

  • Lemma 3.1: KaweckiSmears20SS14
  • Theorem 3.2: KaweckiSmears20SS14
  • Remark 3.1: Simplifications for $C^0$-IP methods
  • Theorem 3.3: KaweckiSmears20
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Lemma 4.1: DominincusGaspozKreuzer19MorinSiebertVeeser08
  • Lemma 4.2: DominincusGaspozKreuzer19MorinSiebertVeeser08
  • Lemma 4.3
  • ...and 47 more