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A curious property of oscillatory FEM solutions of one-dimensional convection-diffusion problems

Madden, Niall, Stynes, Martin (2012)

Applications of Mathematics 2012

Song, Yin and Zhang (Int. J. Numer. Anal. Model. 4: 127-140, 2007) discovered a remarkable property of oscillatory finite element solutions of one-dimensional convection-diffusion problems that leads to a novel numerical method for the solution of such problems. In the present paper this property is described using several figures, then a simple proof of the phenomenon is given which is much more intuitive than the technical analysis of Song et al.

A direct solver for finite element matrices requiring O ( N log N ) memory places

Vejchodský, Tomáš (2013)

Applications of Mathematics 2013

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

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