A DGFEM for nondivergence form elliptic equations with Cordes coefficients on curved domains

Kawecki, Ellya L.

doi:10.1002/num.22372

Cited by 17 publications

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References 32 publications

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“…We have extended the framework introduced in [38], and [24] allowing for domains with curved boundaries, as well as oblique boundary conditions. In doing so, we have introduced a new DGFEM for elliptic equations in nondivergence form, that satisfy the Cordes condition.…”

Section: Resultsmentioning

confidence: 99%

“…In this section, we test the robustness of the scheme (4.47), with the computational domain Ω taken to be the unit disk, approximated in the same manner as in present in [24], Section 3.4. We consider various elliptic operators, L, that satisfy the Cordes condition (3.1).…”

Section: Methodsmentioning

confidence: 99%

“…In the problems we consider, however, there does not appear to be any explicit relationship between the constant and the right hand side, in general. Our approach extends the framework of [38] and [24] (the first of which applies to the Dirichlet boundary condition on polytopal domains, and the second applies to the Dirichlet boundary condition on curved, uniformly convex domains) to the oblique case. To the author's knowledge, there is a sparse amount of work on finite element methods for oblique boundary-value problems present in the existing literature, and, as such, this paper provides the first DGFEM for the approximation of solutions to (1.1).…”

Section: Introductionmentioning

confidence: 99%

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A Discontinuous Galerkin Finite Element Method fOR Uniformly Elliptic Two Dimensional Oblique Boundary-Value Problems

Kawecki¹

2019

SIAM J. Numer. Anal.

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In this paper we present and analyse a discontinuous Galerkin finite element method (DGFEM) for the approximation of solutions to elliptic partial differential equations in nondivergence form, with oblique boundary conditions, on curved domains. In "E. Kawecki, A DGFEM for Nondivergence Form Elliptic Equations with Cordes Coefficients on Curved Domains", the author introduced a DGFEM for the approximation of solutions to elliptic partial differential equations in nondivergence form, with Dirichlet boundary conditions. In this paper, we extend the framework further, allowing for the oblique boundary condition. The method also provides an approximation for the constant occurring in the compatibility condition for the elliptic problems under consideration. * EK acknowledges support of the Engineering and Physical Sciences Research Council [EP/L015811/1].That said, the degenerate (or tangential) oblique problem (falling into the class of degenerate elliptic problems), arises naturally in the (geodetic) problem of determining the gravitational fields of celestial bodies [32]. This problem was discovered by Poincaré [34] during his work on the theory of tides. In the case that d = 2, the oblique boundary-value problem arises in systems of conservation laws in [44,11], where the latter focuses on a mixed elliptic-hyperbolic problem that requires the boundary condition to be strictly oblique. For an overview of the case d = 2 one should refer to [27], and for the case d ≥ 3, one should seek [28]. A particular, and broad subclass of the oblique boundary-value problem is the case when β ≡ n ∂Ω , which is in fact the Neumann boundary-value problem.The author's interest in this type of boundary-value problem, stems from applications to fully nonlinear second order elliptic partial differential equations (PDEs). In particular equations of Monge-Ampère (MA) and Hamilton-Jacobi-Bellman (HJB) type. Upon linearising such equations (for instance by the application of Newton's method), one arrives at an infinite sequence of problems of the form (1.1), and as such, the linear theory contained in this paper will be applicable when considering these nonlinear problems. The MA problem arises in areas such as optimal transport and differential geometry, and has been an area of interest, both from an analytical and a numerical computation point of view for many years, see [42, 10, 33] and [8, 31]; while the HJB problem arises in applications to mean field games, engineering, physics, economics, optimal control, and finance [18,26], where [39,23] mark recent developments in the numerical analysis of such problems.It is clear that the linearisation of HJB and MA type equations results in a sequence of nondivergence form elliptic equations. What is not immediately clear, is how the oblique boundary condition may also arise. In the applications outlined above (geodetic problems, and conservation laws), the oblique boundary condition arises, but these problems are not typically cast in nondivergence form.The nondivergence form oblique boundar...

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Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Discontinuous Galerkin Finite Element Method fOR Uniformly Elliptic Two Dimensional Oblique Boundary-Value Problems

Kawecki¹

2019

SIAM J. Numer. Anal.

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“…Furthermore, exact domain approximation is useful in the design of numerical methods. For example, in the works [17][18][19] a discrete analogue of the Miranda-Talenti estimate [9] motivates the design of the methods, and due to this, the exact approximation of the domain is key to the stability of the methods.…”

Section: Introductionmentioning

confidence: 99%

“…The VEM generally requires that any bilinear forms can be calculated exactly, using the degrees of freedom of the approximation space. It is not clear in the literature that such formulations are extendable to nondivergence form elliptic equations with L ∞ (Ω) coefficients (whereas the results of the current paper are shown to be of benefit to [17, 18]); in particular, the discrete spaces are typically augmented by (theoretically) solving the PDE locally on the physical elements. This consideration is likely to be nontrivial in the setting of nondivergence form equations.…”

Section: Introductionmentioning

confidence: 99%

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

Kawecki

2020

Numerical Methods Partial

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In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré-Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert-Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp-DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates. KEYWORDS discontinuous Galerkin, finite element method, numerical analysis, partial differential equations 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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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 DGFEM for nondivergence form elliptic equations with Cordes coefficients on curved domains

Cited by 17 publications

References 32 publications

A Discontinuous Galerkin Finite Element Method fOR Uniformly Elliptic Two Dimensional Oblique Boundary-Value Problems

A Discontinuous Galerkin Finite Element Method fOR Uniformly Elliptic Two Dimensional Oblique Boundary-Value Problems

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

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

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