Numerical Solution of Cauchy Problems for Elliptic ... - COMSOL.com
Numerical Solution of Cauchy Problems for Elliptic ... - COMSOL.com
Numerical Solution of Cauchy Problems for Elliptic ... - COMSOL.com
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Presented at the <strong>COMSOL</strong> Multiphysics User's Conference 2005 Boston<br />
Rewrite as an Initial value problem:<br />
( ) (<br />
u 0 a<br />
=<br />
−1<br />
au y<br />
y<br />
− ∂<br />
∂x a ∂<br />
∂x<br />
0<br />
) (<br />
u<br />
au y<br />
)<br />
, 0≤y ≤1,<br />
(<br />
u(x, 0)<br />
u y (x, 0)<br />
) ( )<br />
g<br />
= .<br />
h<br />
The unbounded x-derivative makes the problem ill-posed!<br />
Approximation <strong>of</strong> derivative:<br />
∂<br />
∂x<br />
(<br />
a(u) ∂v )<br />
∂x<br />
≈ ¯D(A ¯DV )<br />
A diagonal matrix, depends on u<br />
¯D bounded differentiation matrix, spectral or otherwise<br />
Resulting stable problem is solved using standard code (ode45) in Matlab.<br />
– Femlab Conference 2005 – 7