Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains

Cockburn, Bernardo; Solano, Manuel

doi:10.1137/100805200

Cited by 51 publications

(48 citation statements)

References 12 publications

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“…In the purely diffusive case, this new approximation of u has been proven to converge with order k + 2 for k ≥ 1 when the domain is polygonal ( [4,6]), and also when it has curved Dirichlet boundary ( [8,9]).…”

Section: Numerical Results: Boundary-value Problemmentioning

confidence: 95%

“…One of the first ideas in this direction was introduced by [5] for the one-dimensional case and then extended to higher space dimensions for pure diffusion [8,9] and convection-diffusion [9] equations. In their work, the mesh does not fit the domain and the distance between the computational domain and the boundary Γ := ∂Ω is of only order O(h), making this method attractive from a computational point of view.…”

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confidence: 99%

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A High Order HDG Method for Curved-Interface Problems Via Approximations from Straight Triangulations

2016

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We propose a novel technique to solve elliptic problems involving a non-polygonal interface/boundary. It is based on a high order hybridizable discontinuous Galerkin method where the mesh does not exactly fit the domain. We first study the case of a curved-boundary value problem with mixed boundary conditions since it is crucial to understand the applicability of the technique to curved interfaces. The Dirichlet data is approximated by using the transferring technique developed in a previous paper. The treatment of the Neumann data is new. We then extend these ideas to curved interfaces. We provide numerical results showing that, in order to obtain optimal high order convergence, it is desirable to construct the computational domain by interpolating the boundary/interface using piecewise linear segments. In this case the distance of the computational domain to the exact boundary is only O(h 2 ).

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Section: Numerical Results: Boundary-value Problemmentioning

confidence: 95%

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confidence: 99%

A High Order HDG Method for Curved-Interface Problems Via Approximations from Straight Triangulations

2016

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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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“…The novelty, and strength, of our approach resides on the use of a high order Hybridizable Discontinuous Galerkin (HDG) method combined with a technique for handling geometries with curved boundaries that preserves the order of accuracy of the method. The result is a robust algorithm that is able to provide high order of accuracy for the approximation of both the flux function and the magnetic field, offers good potential for parallel computations [26,27], and provides the flexibility of handling curved geometries (with or without x-points) relying only on polygonal meshes [28].…”

Section: Introductionmentioning

confidence: 99%