General finite-volume framework for saddle-point problems of various physics. (English) Zbl 1503.65205
The author presents a general finite volume framework for saddle-point problems. One of the main tools used in the framework is the application of the Ostrogradsky-Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. The approach is applied to several problems of various physics, namely incompressible elasticity problem, incompressible Navier-Stokes, Brinkman-Hazen-Dupuit-Darcy, Biot, and Maxwell equations. The framework is tested on simple analytical solutions. The article is interesting and it merits to read.
Reviewer: Abdallah Bradji (Annaba)
MSC:
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
| 74S10 | Finite volume methods applied to problems in solid mechanics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76S05 | Flows in porous media; filtration; seepage |
| 78M12 | Finite volume methods, finite integration techniques applied to problems in optics and electromagnetic theory |
| 78A25 | Electromagnetic theory (general) |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
finite volume method; inf-sup stability; harmonic averaging point; multi-physics; Brinkman-Darcy equations; Navier-Stokes equations; Maxwell equations; Biot equationReferences:
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