The expression of Heat2D_VaryingCoef

[DUCH714]

May I ask the expression of the class Heat2D_VaryingCoef, is it orresponded to B.1.7 2D Heat with Varying Coefficients (Heat2d-VC)?

However, in your code, the expression should be

$$ u_t-a(x,y)(\nabla \cdot \nabla u)=f(x), $$

but in B.1.7, the expression is

$$ u_t-\nabla (a(x,y) \nabla u)=f(x). $$

Which one is correct?

[i207M]

Thank you for bringing up this problem. This indeed needs some clarification.

In the case we provided, a piecewise constant $a$ is used. Hence the order of $a$ and $\nabla$ doesn't matter (ignoring the boundary points). We moved $a$ outside to accelerate computing. However, if one wants to update the settings of the problem, one needs to update the implementation.

I hope this answers your question.

[DUCH714]

Thanks for answering the question.

In my opinion, piecewise constant is not constant. If $a$ is piecewise constant on the computational domain, you can't move it outside because the derivative at the discontinous points is not 0.

Especially in numerical methods, the discontinous points may cause oscillations if you calculate the derivative near them. So may I ask when you generated the numerical solution $u$, did you move $a$ outside?

[i207M]

Thanks for your comment. It is debatable whether that will cause much of a difference.
For your second question, we apply the nearest interpolation of $a$ when using the COMSOL solver, with $a$ inside.