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### A direct solver for finite element matrices requiring $O\left(NlogN\right)$ memory places

Applications of Mathematics 2013

We present a method that in certain sense stores the inverse of the stiffness matrix in $O\left(NlogN\right)$ memory places, where $N$ is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires $O\left({N}^{3/2}\right)$ arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with $O\left(NlogN\right)$ operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular domains,...

### A method to rigorously enclose eigenpairs of complex interval matrices

Applications of Mathematics 2013

In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair $x=\left(\lambda ,\right)$ is found by solving a nonlinear equation of the form $f\left(x\right)=0$ via a contraction argument. The set-up of the method relies on the notion of $radiipolynomials$, which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable.

### A parallel method for population balance equations based on the method of characteristics

Applications of Mathematics 2013

In this paper, we present a parallel scheme to solve the population balance equations based on the method of characteristics and the finite element discretization. The application of the method of characteristics transform the higher dimensional population balance equation into a series of lower dimensional convection-diffusion-reaction equations which can be solved in a parallel way. Some numerical results are presented to show the accuracy and efficiency.

### A short philosophical note on the origin of smoothed aggregations

Applications of Mathematics 2013

We derive the smoothed aggregation two-level method from the variational objective to minimize the final error after finishing the entire iteration. This contrasts to a standard variational two-level method, where the coarse-grid correction vector is chosen to minimize the error after coarse-grid correction procedure, which represents merely an intermediate stage of computing. Thus, we enforce the global minimization of the error. The method with smoothed prolongator is thus interpreted as a qualitatively...