Adaptive <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e342" altimg="si3.svg"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math> interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

Brenner, Susanne C.; Kawecki, Ellya L.

doi:10.1016/j.cam.2020.113241

Cited by 15 publications

(22 citation statements)

References 27 publications

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“…Our second main contribution is a proof of reliability and local efficiency of residual-based error estimators in 2 -norms for piecewise polynomial approximations on simplicial meshes, which consist of unweighted volume residuals with appropriately penalized jumps of function gradients and jumps of function values. This extends earlier results for 2 -conforming and 0 -IP methods from [3,9,26]. In fact, owing to the strong solution of the PDE, we show that the a posteriori error analysis is determined primarily by the choice of approximation space and is otherwise independent of the numerical method, so that our a posteriori error bounds applies to any piecewise polynomial function over the mesh.…”

Section: Introductionsupporting

confidence: 85%

“…A posteriori error bounds of a similar nature have been shown already in [3,9,26] for various numerical methods. However, Theorem 4.2 shows that the a posteriori error bounds are not restricted to any particular numerical method, as the bounds apply to arbitrary piecewise polynomial approximations on .…”

Section: Remark 43supporting

confidence: 59%

“…Choosing = 0, = 0 and = 1/2, we obtain the original DGFEM proposed in [54,55], see Lemma 5.1 concerning the equivalence of the stabilization terms. If we take = 1, and = = 0, then we obtain the 0 -interior penalty FEM proposed in [48], and further analysed in [9]. Methods using = 1 are of interest in the context of adaptive methods, see [38].…”

Section: Application To a Family Of Numerical Methodsmentioning

confidence: 99%

“…These results were extended to parabolic problems in [56]. There has since been significant recent activity centred on this approach, including preconditioners [53], adaptive 2 -conforming and mixed methods in [26,28], extensions to curved domains [36], boundary conditions involving oblique derivatives [27,37], and 0 interior penalty (IP) methods [3,9,48].…”

Section: Introductionmentioning

confidence: 99%

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

Kawecki

Smears

2021

ESAIM: M2AN

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We provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and $C^0$-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 $H^2$ 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.

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Section: Introductionsupporting

confidence: 85%

Section: Remark 43supporting

confidence: 59%

Section: Application To a Family Of Numerical Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kawecki

Smears

2021

ESAIM: M2AN

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“…Such problems play a central role in financial mathematics, for example, the valuation of financial products. A closely connected area is the numerical solution of second-order Hamilton-Jacobi-Bellman (HJB) equations [6,7], where the existence of an operator in nondivergence form also follows due to the stochastic influence. In addition to the nonvariational nature of the linear operator, these problems possess further numerical challenges due to nonlinearities introduced by a pointwise minimization.…”

Section: Introductionmentioning

confidence: 99%

Error estimation for second‐order partial differential equations in nonvariational form

Blechschmidt

Herzog

Winkler

2020

Numerical Methods Partial

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Second-order partial differential equations (PDEs) in nondivergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second-order PDEs. The nondivergence form in these problems is natural. If the coefficients of the diffusion matrix are not differentiable, the problem cannot be transformed into the more convenient variational form. We investigate tailored nonconforming finite element approximations of second-order PDEs in nondivergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation. We study both approximations with continuous and discontinuous trial functions. Of particular interest are a priori and a posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature. KEYWORDS finite element error estimates, PDEs in nondivergence form This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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A C⁰ linear finite element method for a second‐order elliptic equation in non‐divergence form with Cordes coefficients

Lin

Zou

2022

Numerical Methods Partial

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In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and a Hessian recovery based linear FEM for second‐order elliptic equations in non‐divergence form. The elliptic equation is casted into a symmetric non‐divergence weak formulation, in which second‐order derivatives of the unknown function are involved. We use gradient and Hessian recovery operators to calculate the second‐order derivatives of linear finite element approximations. Although, thanks to low degrees of freedom of linear elements, the implementation of the proposed schemes is easy and straightforward, the performances of the methods are competitive. The unique solvability and the H2$$ {H}^2 $$ seminorm error estimate of the GRBL scheme are rigorously proved. Optimal error estimates in both the L2$$ {L}^2 $$ norm and the H1$$ {H}^1 $$ seminorm have been proved when the coefficient is diagonal, which have been confirmed by numerical experiments. Superconvergence in errors has also been observed. Moreover, our methods can handle computational domains with curved boundaries without loss of accuracy from approximation of boundaries. Finally, the proposed numerical methods have been successfully applied to solve fully nonlinear Monge–Ampère equations.

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Adaptive C0 interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

Cited by 15 publications

References 27 publications

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

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

Error estimation for second‐order partial differential equations in nonvariational form

A C⁰ linear finite element method for a second‐order elliptic equation in non‐divergence form with Cordes coefficients

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Adaptive C0 interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

Cited by 15 publications

References 27 publications

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

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

Error estimation for second‐order partial differential equations in nonvariational form

A C0 linear finite element method for a second‐order elliptic equation in non‐divergence form with Cordes coefficients

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

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

A C⁰ linear finite element method for a second‐order elliptic equation in non‐divergence form with Cordes coefficients