Virtual enriching operators

Brenner, Susanne C.; Sung, Li-yeng

doi:10.1007/s10092-019-0338-z

Cited by 17 publications

(28 citation statements)

References 24 publications

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“…The analysis hinges on the following Lemma, which shows that the jump seminorm |•| , defined in (3.5) controls the distance of functions from 2 (Ω) ∩ 1 0 (Ω). See also [53] for related results that are explicit in the polynomial degree on more general meshes, and see also the concluding remarks in [10].…”

Section: A Posteriori Error Boundmentioning

confidence: 99%

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: A Posteriori Error Boundmentioning

confidence: 99%

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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“…Remark 4.5 (A comment on conforming error e c ). It is possible to derive an error estimate by splitting the error using an averaging operator to C 1 -conforming macro element or virtual element spaces, as in [16,17,27]. However, such an estimate for the nonconforming part of the estimator requires using L ∞ to L 2 norm polynomial inverse inequalities several times.…”

Section: Lemma 42 (Helmholtz Decomposition)mentioning

confidence: 99%

“…An alternative approach, recently proposed in [35] in the context of nonlinear PDEs in nondivergence form, is to reconstruct the solution into C 1 -conforming spaces introduced in [17,45]. While this allows us to avoid problems with element geometries, it introduces the disadvantage in the current context that the resulting error estimate would gain an additional suboptimality of order p d in d spatial dimensions, due to the repeated application of a polynomial inverse estimate apparently necessary for the analysis.…”

Section: Introductionmentioning

confidence: 99%

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

Dong¹,

Mascotto²,

Sutton³

2021

SIAM J. Numer. Anal.

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We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C 1 -conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H 2 space and a generalised Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

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“…A proof of (29) in the two-dimensional case can be found in [34,Section 4.11.3], [35, Equation (2.9)] for the case p = 2 and in [36, Lemma 3.1] for p ≥ 2. For the three-dimensional case we refer to [30], where a 3D HCT element for polynomial degrees p ∈ {2, 3} is studied and to [37], where a different function space based on virtual elements of arbitrary order is used.…”

Section: Proofmentioning

confidence: 99%

“…which is due to the fact that ∇ 2 h u h ∈  DG h (R × ) holds. The discrete Miranda-Talenti estimates follow after inserting (34) and (36) in case of H = H CG , or (35) and (37) in case of H = H DG , into (33), and combining the resulting estimates with (30) and (31). ▪…”

Section: Proofmentioning

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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Virtual enriching operators

Abstract: We construct bounded linear operators that map H 1 conforming Lagrange finite element spaces to H 2 conforming virtual element spaces in two and three dimensions. These operators are useful for the analysis of nonstandard finite element methods.

Cited by 17 publications

References 24 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

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

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

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Virtual enriching operators

Abstract: We construct bounded linear operators that map H 1 conforming Lagrange finite element spaces to H 2 conforming virtual element spaces in two and three dimensions. These operators are useful for the analysis of nonstandard finite element methods.

Cited by 17 publications

References 24 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

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

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

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Product

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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