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PDE question

Anonymous Sat 07 Nov 12:24:48 2020 No.12314508 View Reply Original Report

Quoted By: >>12315104 >>12315119

I have a boundary value problem for 3D diffusion equation div(D gradC) = 0.

The question is: how do I analyze changes in C when D changes? Is there a theory for this? In which norms do I compare solutions for different D?

Anonymous

Anonymous Sat 07 Nov 2020 17:14:55 No.12315104 Report

Quoted By:

>>12314508
bump

Anonymous

Anonymous Sat 07 Nov 2020 17:20:22 No.12315119 Report

Quoted By:

>>12314508
Yeah I love these kinds of exercise, good luck

Anonymous

Anonymous Sat 07 Nov 2020 17:35:26 No.12315159 Report

Quoted By: >>12315225 >>12317685

Not sure to understand the question.
If D is uniform, the solution is D-independant.

Anonymous

Anonymous Sat 07 Nov 2020 18:13:36 No.12315225 Report

Quoted By: >>12317202

>>12315159
I think what he means is that is there an analytical way to solve for C(D).
My first guess would be helmholtz decomposition and hoping something falls out, but I don't think there's a general way to do it without numerics.

Anonymous

Anonymous Sun 08 Nov 2020 06:49:12 No.12317202 Report

Quoted By: >>12317844

>>12315225
H, I didn't think in a way that my question simply requires analytical solution. Finding which is, however, a practically impossible task for real-world problems which have complex and heterogeneous domains and where D is in fact a s.p.d. matrix.

Initially, I'm trying to solve div(D gradC) = 0 by approaching it through series of problems div(D_n gradC_n) = 0, where D_n -> D, and want to know how C_n -> C

Anonymous

Anonymous Sun 08 Nov 2020 11:24:24 No.12317685 Report

Quoted By:

>>12315159
In my case D is in general a 3x3 matrix, although usually in form diag{Dx,Dy,Dz}. And Neumann BCs are usually defined for -D*gradC, so I can't ignore D

My guess is that my question indeed requires some analytical solution in order to estimate the difference for different D. That, however, can be done only for simple domains and probably for simple D's (not full 3x3 matrix), but some info from that case will be useful too

Anonymous

Anonymous Sun 08 Nov 2020 13:07:17 No.12317844 Report

Quoted By:

>>12317202
In one dimension the problem is characterized by the Peclet number.
Maybe you can start from there.

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