General finite-volume framework for saddle-point problems of various physics

This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies 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. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.

Variational linearization of pure traction problems in incompressible elasticity

We consider pure traction problems, and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.

Preconditioned Conjugate Gradient Methods for the Refined FEM Discretizations of Nearly Incompressible Elasticity Problems in Three Dimensions

Nearly incompressible problems in three dimensions are the important problems in practical engineering computation. The volume-locking phenomenon will appear when the commonly used finite elements such as linear elements are applied to the solution of these problems. There are many efficient approaches to overcome this locking phenomenon, one of which is the higher-order conforming finite element method. However, we often use the lower-order nonconforming elements as Wilson elements by considering the computational complexity for three-dimensional (3D) problems considered. In general, the convergence of Wilson elements will heavily rely on the quality of the meshes. It will greatly deteriorate or no longer converge when the mesh distortion is very large. In this paper, the refined element method based on Wilson element is first applied to solve nearly incompressible elasticity problems, and the influence of mesh quality on the refined element is tested numerically. Its validity is verified by some numerical examples. By using the internal condensation method, the refined element discrete system of equations is deduced into the one which is spectrally equivalent to an 8-node hexahedral element discrete system of equations. And then, a type of efficient algebraic multigrid (AMG) preconditioner is presented by combining both the coarsening techniques based on the distance matrix and the effective smoothing operators. The resulting preconditioned conjugate gradient (PCG) method is efficient for 3D nearly incompressible problems. The numerical results verify the efficiency and robustness of the proposed method.