By Raphael Salem
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Additional info for Algebraic numbers and Fourier analysis
It is convenient to assume that " Ä 1. 2) The function plots for " D 1, " D 0:1, and " D 0:01 follow in Fig. 1. It is clear that when " approaches zero, there is the onset of a boundary layer close to x D 1. 0 u would solve the resulting equation. 1) a first order equation with two boundary conditions, and such problem does not have solutions in general. The limiting equation imposes that u is a constant, but from the boundary conditions such constant has to be equal to one at x D 0, and to zero at x D 1, a clear impossibility.
V in gil C V ex ge /V dx: Since W0 is the argument that minimizes I0 . 15) In Fig. 3 we present a numerical simulation, for N i D N e D 255. Also " D 10 2 , V in D 10, V ex D 65, gil D 4 10 2 , gel D 10 2 . The synapses are disposed periodically, in an alternate fashion. Again, the exact solution is plotted in a solid line. The values of the MsFEM solution are indicated by , and the nodal values of the classical Galerkin solution are plotted with . 5 Conclusions 6 5 Membrane Potential(mV) Fig. 3 Numerical test in the case of large number of synapses.
0 After all, it would be just perfect to have a method that converges (with ") to the correct solution for a fixed mesh. This is not happening. xj /. Assume N even. At the " D 0 limit, ujC1 D uj 1 . This and the boundary conditions originate the oscillatory behavior of the approximate solution. See Fig. 3. 8), this scheme is also a finite difference scheme which uses a central difference approximation for the convective term du=dx. 8 z Fig. 3) for " D 10 5 and N D 16. 2 0 Fig. 9) for the convection term, for " D 0:01 and h D 1=32.
Algebraic numbers and Fourier analysis by Raphael Salem