We have developed easy to use fast multipole method (FMM) libraries for the Laplace, low-frequency Helmholtz, and Stokes equations in two and three dimensions. The codes are based on a new method for applying translation operators and provide reasonable performance on either single core processors, or small multi-core systems using OpenMP.

An artificial boundary condition method, derived in terms of infinite Fourier series, is applied to solve a class of quasi-Newtonian Stokes flows. Based on the natural boundary reduction involving an artificial condition on the artificial boundary, the coupled variational problem and its numerical solution are obtained. The unique solvability of the continuous and discrete formulations are discussed, and the error analysis for the problem is also considered. Finally, an a posteriori error estimate for the corresponding problem is provided.

Cancer has recently overtaken heart disease as the world’s biggest killer. Cancer isinitiated by gene mutations that result in local proliferation of abnormal cells and theirmigration to other parts of the human body, a process called metastasis. The metastasizedcancer cells then interfere with the normal functions of the body, eventually leading todeath. There are two hundred types of cancer, classified by their point of origin. Most ofthem share some common features, but they also have their specific character. In thisarticle we review mathematical models of such common features and then proceed to describemodels of specific cancer diseases.

Cavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models. Herein, we consider the problem of computing the effective permeability of rock samples based on high-resolution 3D CT scans containing millions of voxels. We use the Stokes-Brinkman equations in the entire domain, covering regions of free flow governed by the Stokes equations, porous Darcy flow, and transitions between them. The presence of different length scales and large (ten orders of magnitude) contrasts in permeability leads to highly ill-conditioned linear systems of equations, which are difficult to solve. To obtain a problem that is computationally tractable, we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models. We find that, in terms of effective permeability, the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability. All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows, depending on if there do or do not exist percolating free-flow regions. The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.