A spectral method for an elliptic equation with a nonlinear Neumann boundary condition

Abstract

Let Ω be an open region in ℝ^d, d ≥ 2, that is diffeomorphic to 𝔹^d. Consider solving −Δu + γu = 0 on Ω with the Neumann boundary condition $\frac {\partial u}{\partial \mathbf {n}}=b\left (\cdot ,u\right )$ over ∂Ω. The function b is a nonlinear function of u. The problem is reformulated in a weak form, and then a spectral Galerkin method is used to create a sequence of finite dimensional nonlinear problems. An error analysis shows that under suitable assumptions, the solutions of the finite dimensional problems converge to those of the original problem. To carry out the error analysis, the original problem and the spectral method is converted to a nonlinear integral equation over H^1/2 (Ω) , and the reformulation is analyzed using tools for solving nonlinear integral equations. Numerical examples are given to illustrate the method. In our error analysis, we assume the existence and local uniqueness of a solution. For the case of three dimensions and a nonlinearity b that is given by the Stefan–Boltzmann law, we will provide an existence proof in the final section.

Appendix

Defining surface normals and Jacobian for a general surface. This is well-known in the literature, but we include it for convenience. For notational simplicity in the mapping ${\Phi }:\overline {\mathbb {B}}^{d}\underset {onto}{\overset {1-1}{\longrightarrow }}\overline {{\Omega }}$, let x be replaced by (x, y, z), and s be replaced by (s, t, u). Write

$$\begin{array} [c]{l} s=s(x,y,z)\\ t=t(x,y,z)\\ u=u(x,y,z) \end{array} $$

For derivatives, use the shorthand notation

$$ds_{1}=\frac{\partial s(x,y,z)}{\partial x}\equiv\frac{\partial{\Phi}_{1}(x,y,z)}{\partial x},\quad ds_{2}=\frac{\partial s(x,y,z)}{\partial y},\quad ds_{3}=\frac{\partial s(x,y,z)}{\partial z} $$

with similar notation for t and u.

For the surface Jacobian |J_bdy (x, y, z)| used in the change of variables expression (2.11),

$$\left\vert J_{\textit{bdy}}\left( x,y,z\right) \right\vert^{2}=\left\vert \begin{array} [c]{ccc} x & y & z\\ dt_{1} & dt_{2} & dt_{3}\\ du_{1} & du_{2} & du_{3} \end{array} \right\vert^{2}+\left\vert \begin{array} [c]{ccc} ds_{1} & ds_{2} & ds_{3}\\ x & y & z\\ du_{1} & du_{2} & du_{3} \end{array} \right\vert^{2}+\left\vert \begin{array} [c]{ccc} ds_{1} & ds_{2} & ds_{3}\\ dt_{1} & dt_{2} & dt_{3}\\ x & y & z \end{array} \right\vert^{2} $$

The normal at (s, t, u) = Φ (x, y, z), call it N (s, t, u), is given by

$$\begin{array}{@{}rcl@{}} \mathbf{N} & =&\frac{\mathbf{G}}{\left\Vert \mathbf{G}\right\Vert },\\ \mathbf{G} & =&\left[ \begin{array} [c]{c} \left( dt_{1}du_{2}-dt_{2}du_{1}\right) z+\left( dt_{3}du_{1}-dt_{1} du_{3}\right) y+\left( dt_{2}du_{3}-dt_{3}du_{2}\right) x\\ \left( du_{1}ds_{2}-du_{2}ds_{1}\right) z+\left( du_{3}ds_{1}-du_{1} ds_{3}\right) y+\left( du_{2}ds_{3}-du_{3}ds_{2}\right) x\\ \left( ds_{1}dt_{2}-ds_{2}dt_{1}\right) z+\left( ds_{3}dt_{1}-ds_{1} dt_{3}\right) y+\left( ds_{2}dt_{3}-ds_{3}dt_{2}\right) x \end{array} \right] . \end{array} $$

As an example, consider the ellipsoidal mapping

$${\Phi}\left( x,y,z\right) =\left( ax,by,cz\right) ,\quad\quad\left( x,y,z\right) \in\overline{\mathbb{B}}^{3}. $$

Then

$$\left\vert J_{\textit{bdy}}\left( x,y,z\right) \right\vert^{2}=\left( bcx\right)^{2}+\left( acy\right)^{2}+\left( abz\right)^{2},\quad \quad\left( x,y,z\right) \in\mathbb{S}^{2}. $$