A hybrid finite volume – finite element method for bulk–surface coupled problems

Chernyshenko, Alexey Y.; Olshanskii, Maxim A.; Vassilevski, Yuri

doi:10.1016/j.jcp.2017.09.064

Cited by 20 publications

(20 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If Γ i is a planar component, then numerical integration is straightforward. For a curvilinear Γ, in general, we need to know a (local) parametrization of the surface to compute integrals in (12). For implicitly given surfaces (for example, for surfaces defined as the zero of a distance function), the numerical integration is a more subtle issue; see, e.g., [34].…”

Section: Numerical Integrationmentioning

confidence: 99%

“…Application of geometrically unfitted finite element methods for the modelling of flow and transport in fractured porous medium have been addressed recently in a number of publications; see, e.g., [5,17,27]. Developments most closely related to the approach taken in the present paper are those found in [12,18,25]. Thus in [18] the authors consider a low order Raviart-Thomas finite element method for the Darcy flow on a 1D network of fractures.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An unfitted finite element method for the Darcy problem in a fracture network

Chernyshenko

Olshanskii

2020

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

The paper develops an unfitted finite element method for solving the Darcy system of equations posed in a network of fractures embedded in a porous matrix. The approach builds on the Hughes-Masud stabilized formulation of the Darcy problem and the trace finite element method. The system of fractures is allowed to cut through the background mesh in an arbitrary way. Moreover, the fractures are not triangulated in the common sense and the junctions of fractures are not fitted by the mesh. To couple the flow variables at multiple fracture junctions, we extend the Hughes-Masud formulation by including penalty terms to handle interface conditions. One observation made here is that by over-penalizing the pressure continuity interface condition one can avoid including additional jump terms along the fracture junctions. This simplifies the formulation while ensuring the optimal convergence order of the method. The application of the trace finite element allows to treat both planar and curvilinear fractures with the same ease. The paper presents convergence analysis and assesses the performance of the method in a series of numerical experiments. For the background mesh we use an octree grid with cubic cells. The flow in the fracture can be easily coupled with the flow in matrix, but we do not pursue the topic of discretizing such coupled system here.

show abstract

Section: Numerical Integrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An unfitted finite element method for the Darcy problem in a fracture network

Chernyshenko

Olshanskii

2020

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Among the various state-of-the art numerical methods for the spatial discretisation of BSPDEs existing in the literature we mention finite elements [25,31,35,36], trace finite elements [30], cut finite elements [16] and discontinuous Galerkin methods [19]. The purpose of the present paper is to introduce a bulk-surface virtual element method (BSVEM) for the spatial discretisation of a coupled system of BSPDEs in two space dimensions.…”

Section: Introductionmentioning

confidence: 99%

Bulk-surface virtual element method for systems of PDEs in two-space dimension

Frittelli,

Madzvamuse,

Sgura

2020

Preprint

View full text Add to dashboard Cite

In this manuscript we present a novel bulk-surface virtual element method (BSVEM) for the numerical approximation of coupled systems of elliptic and parabolic bulk-surface partial differential equations (BSPDEs) in two dimensions. To the best of the authors' knowledge, the proposed method is the first application of the virtual element method [5] to BSPDEs. The BSVEM is based on the discretisation of the bulk domain into polygonal elements with arbitrarily many edges, rather than just triangles. The polygonal approximation of the bulk induces a piecewise linear approximation of the one-dimensional surface (a curve). The bulk-surface finite element method on triangular meshes [25] is a special case of the proposed method. The present work contains several contributions. First, we show that the proposed method has optimal second-order convergence in space, provided the exact solution is H 2+1/4 in the bulk and H 2 on the surface, where the additional 1 4 is required by the combined effect of surface curvature and polygonal elements. Our analysis shows, as a byproduct, that first-degree virtual elements for bulk-only PDEs retain optimal convergence in the presence of surface curvature. In carrying out the analysis we provide novel theoretical tools for the analysis of curved boundaries and non-constant boundary conditions, such as the Sobolev extension and a special inverse trace operator. We show that general polygons can be exploited to reduce the computational complexity of the matrix assembly. Moreover, we present an optimised matrix implementation that can also be exploited in the pre-existing special case of bulk-surface finite elements on triangular meshes [25]. Numerical examples illustrate our findings and experimentally show the optimal convergence rates in space and time.

show abstract

“…Strategies to enhance FE computations by coupling them with other discretization methods have received a lot of attention both in the mathematical and engineering community. Examples of successful couplings include, e.g., FE and meshless methods, [1][2][3][4] FE and finite volume methods, [5][6][7] as well as advanced discretization techniques as mixed and discontinuous Galerkin (DG) methods. 8,9 The idea of coupling distinct numerical techniques in different regions of the computational domain may also be interpreted in the framework of domain decomposition (DD) methods.…”

Section: Introductionmentioning

confidence: 99%

Hybrid coupling of CG and HDG discretizations based on Nitsche’s method

2019

View full text Add to dashboard Cite

A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal and elastic problems. The hybrid CG-HDG coupling exploits the definition of the numerical flux and the trace of the solution on the mesh faces to impose the transmission conditions between the CG and HDG subdomains. The continuity of the solution is imposed in the CG problem via Nitsche's method, whereas the equilibrium of the flux at the interface is naturally enforced as a Neumann condition in the HDG global problem. The proposed strategy does not affect the core structure of CG and HDG discretizations. In fact, the resulting formulation leads to a minimally-intrusive coupling, suitable to be integrated in existing CG and HDG libraries. Numerical experiments in two and three dimensions show optimal global convergence of the stress and superconvergence of the displacement field, locking-free approximation, as well as the potential to treat structural problems of engineering interest featuring multiple materials with compressible and nearly incompressible behaviors.

show abstract

A hybrid finite volume – finite element method for bulk–surface coupled problems

Cited by 20 publications

References 51 publications

An unfitted finite element method for the Darcy problem in a fracture network

An unfitted finite element method for the Darcy problem in a fracture network

Bulk-surface virtual element method for systems of PDEs in two-space dimension

Hybrid coupling of CG and HDG discretizations based on Nitsche’s method

Contact Info

Product

Resources

About