A new method for numerical solution of checkerboard fields

Abstract

We consider a generalized version of the standard checkerboard and discuss the difficulties of finding the corresponding field by standard numerical treatment. A new numerical method is presented which converges independently of the local conductivities

A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids

“…Using a grid with approximately four times as many elements as the finest grid for the TPS scheme computations in Table 3, we observe a very slight decrease in the accuracy of the approximations for K E,high and K E,vhigh . We note that this somewhat surprising observation is actually in accordance with the findings reported in [43] for a checkerboard problem using a finite element method. It is found in [43] that the finite element method is actually diverging for high permeability contrasts.…”

“…We note that this somewhat surprising observation is actually in accordance with the findings reported in [43] for a checkerboard problem using a finite element method. It is found in [43] that the finite element method is actually diverging for high permeability contrasts. The authors then develop a special finite element method, which can treat this highly challenging problem also in the case of high permeability contrasts.…”

The effective conductivity of arrays of squares: Large random unit cells and extreme contrast ratios

“…All these difficulties add up and may manifest themselves as artificial ill-conditioning, slow convergence with mesh refinement, critical slowing down in iterative solvers, and severe loss of precision. Several methods have been suggested to alleviate these problems including variants of the finite element method [1,6], network models [23,12], renormalization schemes [20], mode-matching methods [27], and Brownian motion simulation [21]. See also Section 3 of [26] for state-of-the-art algorithms to combat critical slowing down in network models and [8] for a discussion of future directions in the research field at large.…”

“…where I is the identity, K is an integral operator which is compact on smooth Γ, µ(z) is an unknown layer density, and g(z) is a right hand side. Solvers for large-scale boundary values problems on smooth domains often rely on integral equation reformulations of the form (1). The last few years have seen increased activity in the development of efficient solvers using (1) also when Γ is non-smooth.…”

The effective conductivity of random checkerboards

“…The internal boundaries meet at quadruple-junctions (four-wedge corners), where strongly singular fields may arise. See Theorem 3.1 and Section 6 of [1] for the difficulties encountered within the framework of the finite element method. Nevertheless, remarkably simple closed-form expressions for the effective conductivity of two-component ordered checkerboards were found a long time ago [7,18,20].…”