# Solving a boundary value problem with variable number of coupled equations

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)

• 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? Feb 27 at 22:14
• @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.
– Masa
Feb 28 at 7:57