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We consider the coupling across an interface of a fluid flow and a porous media flow. The differential equations involve Stokes equations in the fluid region, Darcy equations in the porous region, plus a coupling through an interface with Beaver–Joseph–Saffman transmission conditions. The discretization consists of P2/P1 triangular Taylor–Hood finite elements in the fluid region, the lowest order triangular Raviart–Thomas finite elements in the porous region, and the mortar piecewise constant Lagrange multipliers on the interface. We allow for nonmatching meshes across the interface. Due to the small values of the permeability parameter of the porous medium, the resulting discrete symmetric saddle point system is very ill conditioned. We design and analyze preconditioners based on the finite element by tearing and interconnecting (FETI) and balancing domain decomposition (BDD) methods and derive a condition number estimate of order for the preconditioned operator. In case the fluid discretization is finer than the porous side discretization, we derive a better estimate of order for the FETI preconditioner. Here is the mesh size of the porous side triangulation. The constants and are independent of the permeability , the fluid viscosity , and the mesh ratio across the interface. Numerical experiments confirm the sharpness of the theoretical estimates.
Cells use signaling networks consisting of multiple interacting proteins to respond to changes in their environment. In many situations, such as chemotaxis, spatial and temporal information must be transmitted through the network. Recent computational studies have emphasized the importance of cellular geometry in signal transduction, but have been limited in their ability to accurately represent complex cell morphologies. We present a finite volume method that addresses this problem. Our method uses Cartesian-cut cells in a differential algebraic formulation to handle the complex boundary dynamics encountered in biological systems. The method is second-order in space and time. Several models of signaling systems are simulated in realistic cell morphologies obtained from live cell images. We then examine the effects of geometry on signal transduction.
We propose a family of Markov chain Monte Carlo methods whose performance is unaffected by affine tranformations of space. These algorithms are easy to construct and require little or no additional computational overhead. They should be particularly useful for sampling badly scaled distributions. Computational tests show that the affine invariant methods can be significantly faster than standard MCMC methods on highly skewed distributions.
We present a numerical method for computing the signed distance to a piecewise-smooth surface defined as the zero set of a function. It is based on a marching method by Kim (2001) and a hybrid discretization of first- and second-order discretizations of the signed distance function equation. If the solution is smooth at a point and at all of the points in the domain of dependence of that point, the solution is second-order accurate; otherwise, the method is first-order accurate, and computes the correct entropy solution in the presence of kinks in the initial surface.
Given a two times differentiable curve in the plane, I prove that — using only the volume fractions associated with the curve — one can construct a piecewise linear approximation that is second-order in the max norm. I derive two parameters that depend only on the grid size and the curvature of the curve, respectively. When the maximum curvature in the 3 by 3 block of cells centered on a cell through which the curve passes is less than the first parameter, the approximation in that cell will be second-order. Conversely, if the grid size in this block is greater than the second parameter, the approximation in the center cell can be less than second-order. Thus, this parameter provides an a priori test for when the interface is under-resolved, so that when the interface reconstruction method is coupled to an adaptive mesh refinement algorithm, this parameter may be used to determine when to locally increase the resolution of the grid.