Mimetic Discretizations of Elliptic Control Problems

Antonietti, Paola F.; Bigoni, Nadia; Verani, Marco

doi:10.1007/s10915-012-9659-7

Cited by 23 publications

(18 citation statements)

References 33 publications

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“…We then let the post-processed continuous control be u = u. • hMFD [1]: T is a polygonal/polyhedral mesh, the state and adjoint unknowns (y, p) are approximated using mixed-hybrid mimetic finite differences (hMFD), and the control u is approximated using piecewise constant functions on T . The hMFD schemes form a sub-class of the hybrid mimetic mixed (HMM) methods [17,18] presented in the appendix (Section 7.1); see also Section 6 for corresponding numerical tests.…”

Section: Abbreviationsmentioning

confidence: 99%

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

Droniou¹,

Nataraj²,

Shylaja³

2017

SIAM J. Control Optim.

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In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality system of the control problem. Error estimates for state, adjoint and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution is discussed. These super-convergence results are shown to apply to non-conforming P 1 finite elements, and to the mixed/hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, non-conforming and mixed-hybrid mimetic finite difference schemes.

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Section: Abbreviationsmentioning

confidence: 99%

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

Droniou¹,

Nataraj²,

Shylaja³

2017

SIAM J. Control Optim.

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“…• quasilinear elliptic problems [11], nonlinear and control problems [6,7], obstacle elliptic problem [10],…”

Section: 5mentioning

confidence: 99%

Annotations on the virtual element method for second-order elliptic problems

Manzini

2017

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stiffness matrix is rank deficient). In the VEM we fix this issue by adding a stabilization term to the bilinear form. Stabilization terms are required not to upset the exactness property for polynomials and to scale properly with respect to the mesh size and the problem coefficients.1.4.1. The low-order setting for polygonal cells (in 2D) and polyhedral cells (in 3D) can be obtained as a straightforward generalization of the FEM. The high-order setting deserves special care in the treatment of variable coefficients in order not to loose the optimality of the discretization and in the 3D formulation.1.5. The shape functions are virtual in the sense that we never compute them explicitly (not even approximately) inside the elements. Note that the polynomial projection of the approximate solution is readily available from the degrees of freedom and we can use it as the numerical solution.1.6. The major advantage offered by the VEM is in the great flexibility in admitting unstructured polygonal and polyhedral meshes.1.7. The VEM can be seen as an evolution of the MFD method. Background material (and a few historical notes) will be given in the final section of this document.

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“…The dG method has been recently proven to successfully support polytopic meshes: we refer the reader, e.g., to [7,8,9,10,11,12,13,14,15], as well as to the comprehensive research monograph by Cangiani et al [16]. In addition to the dG method, several other methods are capable to support polytopic meshes, such as the Polygonal Finite Element method [17,18,19,20], the Mimetic Finite Difference method [21,22,23,24], the Virtual Element method [25,26,27,28], the Hybridizable Discontinuous Galerkin method [29,30,31,32,33], and the Hybrid High-Order method [34,35,36,37,38].…”

Section: Introductionmentioning

confidence: 99%

A high-order discontinuous Galerkin approach to the elasto-acoustic problem

Antonietti

Bonaldi

Mazzieri

2020

Computer Methods in Applied Mechanics and Engineering

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We address the spatial discretization of an evolution problem arising from the coupling of viscoelastic and acoustic wave propagation phenomena by employing a discontinuous Galerkin scheme on polygonal and polyhedral meshes. The coupled nature of the problem is ascribed to suitable transmission conditions imposed at the interface between the solid (elastic) and fluid (acoustic) domains. We state and prove a well-posedness result for the strong formulation of the problem, present a stability analysis for the semi-discrete formulation, and finally prove an a priori hp-version error estimate for the resulting formulation in a suitable (mesh-dependent) energy norm. We also discuss the time integration scheme employed to obtain the fully discrete system. The convergence results are validated by numerical experiments carried out in a two-dimensional setting.

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Mimetic Discretizations of Elliptic Control Problems

Cited by 23 publications

References 33 publications

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

Annotations on the virtual element method for second-order elliptic problems

A high-order discontinuous Galerkin approach to the elasto-acoustic problem

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