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How to properly define the divergence of the gradient of a scalar variable manually?
Posted 2025年2月25日 GMT+8 17:25 Modeling Tools & Definitions, Physics Interfaces Version 6.3 4 Replies
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Hi,
I am trying to define the divergence of the gradient of a scalar variable, namely
,
and use it as a source term in a general form PDE in a 2D model, where 
is a constant and 
is the dependent variable.
By the definition of the divergence of the gradient of a scalar variable, I assume what I should do is define a variable in COMSOL like below
Laplacian_xi=d(d(xi,x),x)+d(d(xi,y),y)
which corresponds to
.
However, if I put Laplacian_xi in the source term of the general form PDE, and set Conserved flux as 0, the result differs greatly from defining d(xi,x) and d(xi,y) in the Conserved flux and set the source term as 0.
Can anyone tell me what I did wrong? Do I need to define any weak form to make d(d(xi,x),x)+d(d(xi,y),y) work properly?
Best Runzi Wang
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The most important consideration here is the order of the shape functions used for your dependent variables. If you, for example, would use linear shape functions, the second derivative is =0 by definition. But even with quadratic shape functions, the prediction of second derivatives is bad (element constant).
If you really need a formulation where the Laplacian is used, you need at least third-order shape functions or a very dense mesh to get reasonable accuracy.
-------------------Henrik Sönnerlind
COMSOL
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Hi Henrik, thank you for your reply.
I am wondering what does COMSOL do behind the Conserved flux 
and what are the equations. Because when I put d(xi,x) and d(xi,y) in the Conserved flux, the result seems correct even if I use linear shape function.
If I can understand how does COMSOL solve , then I can try reproduce it manually. But I couldn't find the way COMSOL do it in the equation view. Could you tell me what does COMSOL do or where to find it?
Best
Runzi Wang
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In Equation view, the weak contribution is
g.Gax*test(u3x)+g.Gay*test(u3y)+g.Gaz*test(u3z)
where g.Gax is the input for Conservative Flux. Thus, you have only first derivative in that case when you enter d(xi,x).
-------------------Henrik Sönnerlind
COMSOL
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I see. Thank you for you help.