A Spectral Method for Nonlinear Elliptic Equations

TL;DR

This paper develops a spectral Galerkin method for solving nonlinear elliptic PDEs on smooth bounded domains by transforming them to the unit ball, achieving rapid convergence with smooth data and boundary conditions.

Contribution

It introduces a spectral method for nonlinear elliptic equations using domain transformation and polynomial approximation, demonstrating rapid convergence and extending to Neumann boundary conditions.

Findings

Converges faster than any power of 1/n for smooth problems.

Numerical examples show exponential convergence rates.

Method applicable to both Dirichlet and Neumann boundary conditions.

Abstract

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving an elliptic partial differential equation $Lu=f$ over $\Omega$ with zero Dirichlet boundary value. The function $f$ is a nonlinear function of the solution $u$. The problem is converted to an equivalent\ elliptic problem over the open unit ball $\mathbb{B}^{d}$ in $\mathbb{R}^{d}$, say $\widetilde{L}\widetilde{u}=\widetilde{f}$. Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $\widetilde{u}_{n}$ of degree $\leq n$ that is convergent to $\widetilde{u}$. The transformation from $\Omega$ to $\mathbb{B}^{d}$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\infty}\left( \overline{\Omega }\right) $ and assuming $\partial\Omega$ is a $C^{\infty}$ boundary, the convergence of $\left\Vert \widetilde{u}-\widetilde{u}_{n}\right\Vert _{H^{1}% }$ \ to zero is faster than any power of $1/n$. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to $-\Delta u+\gamma u=f$ with a zero Neumann boundary condition is also presented.