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The Incompressible Flow & Iterative Solver Software (IFISS) package contains software which can be run with MATLAB or Octave to create a computational laboratory for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions, together with state-of-the-art preconditioned iterative solvers for the resulting discrete linear equation systems. In this paper we give a flavor of the code's main features and illustrate its applicability using several case studies. We aim to show that IFISS can be a valuable tool in both teaching and research.

We show that, over the base theory RCA0\\textit {RCA}_0, Stable Ramsey’s Theorem for Pairs implies neither Ramsey’s Theorem for Pairs nor Σ20\\Sigma ^0_2-induction.

Bounds are developed for the condition number of the linear finite element equations of an anisotropic diffusion problem with arbitrary meshes. They depend on three factors. The first factor is proportional to a power of the number of mesh elements and represents the condition number of the linear finite element equations for the Laplacian operator on a uniform mesh. The other two factors arise from the mesh nonuniformity viewed in the Euclidean metric and in the metric defined by the diffusion matrix. The new bounds reveal that the conditioning of the finite element equations with adaptive anisotropic meshes is much better than what is commonly assumed. Diagonal scaling for the linear system and its effects on the conditioning are also studied. It is shown that the Jacobi preconditioning, which is an optimal diagonal scaling for a symmetric positive definite sparse matrix, can eliminate the effects of mesh nonuniformity viewed in the Euclidean metric and reduce those effects of the mesh viewed in the metric defined by the diffusion matrix. Tight bounds on the extreme eigenvalues of the stiffness and mass matrices are obtained. Numerical examples are given.

Two new two-grid algorithms are proposed for solving the Maxwell eigenvalue problem. The new methods are based on the two-grid methodology recently proposed by Xu and Zhou [Math. Comp., 70 (2001), pp. 17–25] and further developed by Hu and Cheng [Math. Comp., 80 (2011), pp. 1287–1301] for elliptic eigenvalue problems. The new two-grid schemes reduce the solution of the Maxwell eigenvalue problem on a fine grid to one linear indefinite Maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid. The new schemes, therefore, save total computational cost. The error estimates reveals that the two-grid methods maintain asymptotically optimal accuracy, and the numerical experiments presented confirm the theoretical results.

The Brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium, where fractures, bubbles, or channels alternate inside a porous matrix. In this work, we explore a novel mixed formulation of the Brinkman problem based on the Hodge decomposition of the vector Laplacian. Introducing the flow's vorticity as an additional unknown, this formulation allows for a uniformly stable and conforming discretization by standard finite elements (Nédélec, Raviart–Thomas, piecewise discontinuous). A priori error estimates for the discretization error in the H(curl; Ω) – H (div; Ω) – L2 (Ω) norm of the solution, which are optimal with respect to the approximation properties of finite element spaces, are obtained. The theoretical results are illustrated with numerical experiments. Finally, the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for H(curl) and H(div) problems available in the preconditioning library hypre (http://llnl.gov/CASC/hypre), as shown in a follow-up paper [P. S. Vassilevski and U. Villa, SIAM J. Sci. Comput., 35 (2013), pp. S3–S17].

Boundary value problems for the Poisson equation in the exterior of an open bounded Lipschitz curve 𝒞 can be recast as first-kind boundary integral equations featuring weakly singular or hypersingular boundary integral operators (BIOs). Based on the recent discovery in [C. Jerez-Hanckes and J. Nédélec, SIAM J. Math. Anal., 44 (2012), pp. 2666–2694] of inverses of these BIOs for 𝒞 = [-1, 1], we pursue operator preconditioning of the linear systems of equations arising from Galerkin–Petrov discretization by means of zeroth- and first-order boundary elements. The preconditioners rely on boundary element spaces defined on dual meshes and they can be shown to perform uniformly well independently of the number of degrees of freedom even for families of locally refined meshes.

We deal with real preconditioned interval linear systems of equations. We present a new operator, which generalizes the interval Gauss–Seidel operator. Also, based on the new operator and properties of well-known methods, we propose a new algorithm, called the magnitude method. We illustrate by numerical examples that our approach outperforms some classical methods with respect to both time and sharpness of enclosures.

Exposure to sublethal heat stress activates a complex cascade of signaling events, such as activators (NO), signal molecules (PKCɛ), and mediators (HSP70 and COX-2), leading to implementation of heat preconditioning, an adaptive mechanism which makes the organism more tolerant to additional stress. We investigated the time frame in which these chemical signals are triggered after heat stress (41 ± 0.5°C/45 min), single or repeated (24 or 72 h after the first one) in heart tissue of male Wistar rats. The animals were allowed to recover 24, 48 or 72 h at room temperature. Single heat stress caused a significant increase of the concentration of HSP70, NO, and PKC level and decrease of COX-2 level 24 h after the heat stress, which in the next course of recovery gradually normalized. The second heat stress, 24 h after the first one, caused a significant reduction of the HSP70 levels, concentration of NO and PKCɛ, and significant increase of COX-2 concentration. The second exposure, 72 h after the first heat stress, caused more expressive changes of HSP70 and NO in the 24 h-recovery groups. The level of PKCɛ was not significantly changed, but there was significantly increased COX-2 concentration during recovery. Serum activity of AST, ALT, and CK was reduced after single exposure and increased after repeated exposure to heat stress, in both time intervals. In conclusion, a longer period of recovery (72 h) between two consecutive sessions of heat stress is necessary to achieve more expressive changes in mediators (HSP70) and triggers (NO) of heat preconditioning.

In this paper we propose and investigate some edge element approximations for three Maxwell systems in three dimensions: the stationary Maxwell equations, the time-harmonic Maxwell equations, and the time-dependent Maxwell equations. These approximations have three novel features. First, the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of finite element meshes, including the stationary Maxwell equations. Second, they ensure the optimal strong convergence of Gauss' laws in some appropriate norm, in addition to the standard optimal convergence in energy norm, under the general weak regularity assumptions that hold for both convex and nonconvex polyhedral domains and for the discontinuous coefficients that may have large jumps across the interfaces between different media. Finally, no saddle-point discrete systems are needed to solve for the stationary Maxwell equations, unlike most existing edge element schemes.

Optimized Schwarz methods are based on transmission conditions between subdomains which are optimized for the problem class that is being solved. Such optimizations have been performed for many different types of partial differential equations, but were almost exclusively based on the assumption of straight interfaces. We study in this paper the influence of curvature on the optimization, and we obtain four interesting new results: first, we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates. Second, we develop an asymptotically accurate approximation technique, based on Turán type inequalities in our case, to solve the much harder optimization problem on the curved interface, and this approximation technique will also be applicable to currently too complex best approximation problems in the area of optimized Schwarz methods. Third, we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis but that these are not the best choices for the performance of the optimized Schwarz method. Finally, we find that in the case of curved interfaces, optimized Schwarz methods are not necessarily convergent for all admissible parameters. Our optimization leads, however, to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition. We illustrate our analysis with numerical experiments.