Convergence of Adaptive Discontinuous Galerkin and $$C^0$$-Interior Penalty Finite Element Methods for Hamilton–Jacobi–Bellman and Isaacs Equations

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 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.

Unified analysis of discontinuous Galerkin andC0-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations

“…We now consider how the abstract framework for analysis in the sections above applies to a family of numerical methods that includes as special cases the methods of [48,54,55] as well as some original methods which are studied further in the context of adaptive methods in [38].…”

“…The function ℎ is uniformly bounded in Ω, and is only defined up to sets of zero ℋ −1 -measure, which is sufficient for our purposes since ℎ only appears in integrals over sets of dimensions − 1 and . The motivation for this particular definition of ℎ can be found in the analysis of adaptive methods, see [38] for further details. For the purposes of this work, it is of course possible to consider common alternative definitions of ℎ that are equivalent up to constants depending on shape-regularity of the mesh.…”

“…Theorem 4.2 shows that the residual-type estimators in (4.4) are reliable and locally efficient. In [38], we use these estimators to construct and prove convergence of adaptive DG and 0 -IP methods for the problem at hand. Note that the estimators in the present setting have some notable differences with residual estimators for approximations of divergence form elliptic problems in 1 -type norms [58].…”

“…The estimators defined in (4.4) do not include any weighting of the volume residual terms with positive powers of the mesh-size function ℎ ; this is indeed both natural and optimal as shown by the efficiency bounds (4.7) and (4.8). This has important ramifications for the analysis of adaptive methods [38]. In comparison to residual estimators for divergence form elliptic problems, here the residual term for the PDE is entirely located on the elements, and the face terms measure only the nonconformity of the approximations.…”

“…We then prove convergence of the numerical approximations in the small-mesh limit for sequences of shape-regular meshes, without any additional regularity assumptions. We then show how our framework applies to a broad family of DG and 0 -IP methods which include as special cases the methods of [54,55] (restricted here to simplicial meshes and fixed polynomial degrees), the method of [48], as well as some original variants that are of further interest in the context of adaptive methods [38]. Thus, up to the constants involved, all of these methods are quasi-optimal and converge in the minimal regularity setting.…”