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Let's assume the equation $$ \nabla^2u_n(\vec{r})+a_n(\vec{r})u_n(\vec{r})=\sum_{m=1}^{N}b_{nm}u_m(\vec{r}),\quad n=1,2,\dots,N,\quad \vec{r}\in\varOmega\tag{1}\label{eq1}, $$ is to be solved for $u_n$ in a domain $\varOmega$ with $$ \hat{e}\cdot\nabla u_n(\vec{r})=0,\quad n=1,2,\dots,N,\quad \vec{r}\in\partial\varOmega^\mathrm{N}, $$ $$ \hat{e}\cdot\nabla u_n(\vec{r})=\sum_{m=1}^{N}c_{nm}u_m(\vec{r}),\quad n=1,2,\dots,N,\quad \vec{r}\in\partial\varOmega^\mathrm{R}, $$ as the boundary conditions, where $a_n$ is a known function, $b_{nm}$ and $c_{nm}$ are known constants, and $\hat{e}$ is the toward normal. The calculated $u_n$ are used to compute the quantity $ U(\vec{r})=\sum_{n}\alpha_nu_n(\vec{r}) $ with $\alpha_n$ being known constants. Therefore, the higher order $N$ used in Eq. (1) the more accurate the quantity $U$ will be.

I wonder if it is possible to solve equations like Eq. (1) with different (variable) orders $N$ in different subdomains of $\varOmega$ using the FEM. (I am able to solve the equation with a uniform order $N$ throughout the domain $\varOmega$ using the FEM)

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  • $\begingroup$ You mean that in some parts of the domain, only $u_1,\ldots,\u_{N_1}$ exist whereas in others you have $u_1,\ldots,\u_{N_2}$? My question would then be what the boundary conditions for those $u_n$ are that live in only part of the domain? $\endgroup$
    – Wolfgang Bangerth
    Feb 27 at 22:14
  • $\begingroup$ @WolfgangBangerth: Yes. In some parts, based on physical analysis and experience, we guess that the order $N_1$ is accurate enough, but in another part, the order $N_2$ should be used. You can think of it as something like $h$- or $p$-refinement in the FEM. I've added the B.C. to the question. For any order $N$, there are $N$ equations with $N$ boundary conditions. $\endgroup$
    – Masa
    Feb 28 at 7:57

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This is a situation where some variables only live on parts of the domain. This is not so different from multiphysics problems -- say, a fluid-structure interaction problem where velocity and pressure variables only live on the fluid part of the domain, and the displacements only live on the solid part of the domain.

To see how something like this can be implemented, you might want to take a look at the step-46 tutorial of the deal.II library. (Disclaimer: I'm both the author of that program and one of the principal authors of the deal.II library.)

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