For example, in the MLS method, the error at the points near the boundary are more than the points in the domain under study. How can we manage the error at points near the boundary?
Masoumeh: one needs to create an adjustable finite elements grid, say parabolic grid with higher density near the boundary, as we used for anisotropic photopolymerization kinetics. In finite differences I used Gear algorithm. You can contact Uriy Krongauz, who is a specialist in adjusting Galerkin methods of PDE solutions. I briefly met him in the past. Sincerely, Vadim
If the solution is smooth, then chebyshev spectral method would be fine. You should get spectral accuracy. If you believe there might be high oscillation in solution at the boundary then spectral method might have troubles. I'm assuming you have Neumann or robin boundary conditions.
Boundary element method might be of interest as well - since you are focused on getting the boundary right.
Try with MATLAB having Chebyshev polynomial .Contact H.J.Korsch .
Best wishes
B.Rath
There is a lot of numerical methods for solving boundary/boundary-initial problems (described by the PDE): boundary element method, finite element method, finite difference method, finite volume method, meshless methods (GFDM), etc.
The choice of method depends on many aspects: task type (eg. thermal, strength of materials,...), computing performance, mathematical difficulties.
On the other hand, each method has its variants, eg. FDM can be implicit/explicity.
Masoumeh, you should give more information about your question.
I liked adaptive Gear algorithm in finite differences method for fast changing boundary parameters. It worked for combination of fast radical reactions and slow molecular transport into reaction zone. Solution instability was minimal. Again, ask Uri Krongauz
Dear all,
I could not find Uri Krongauz 's page.
Actually, I am trying to solve a system of PDE by the Moving least squares method .
in generally with the boundary conditions
B (u,du/dn)=0
or in the one-dimensional time-dependent problems with initial value
Dear Masoumeh
Could you explain, why you are so interested in very precise values on boundary if you have such simple boundary condition?
In my opinion, in this case, most methods will be sufficiently accurate. But still if you want very precise boundary solution, then you should use eg. general boundary element method. There is a lot of literature in this topic.
But, in my opinion, one of the simplest method (in terms of mathematics) is a finite difference method. You can improve of accuracy by using the high order derivatives, look here: http://amcm.pcz.pl/get.php?article=2012_1/art_07.pdf
Best regards
Lukasz
http://amcm.pcz.pl/get.php?article=2012_1/art_07.pdf
Increasing the mesh at the boundary should be a good solution but that will cost the computation
I think a lot of methods solve the PDE but the best one, are depend on the problems
i. .e depend on the mesh grid points, so the finite element methods has many degree Poly. to solve PDE, you can try to get the best method should be apply on the problems
Hi
See the work of L.N.Trefethen in MATLAB. Hope this will help you solve your problems
Prof B.Rath
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