An Accelerated Method for Nonlinear Elliptic PDE

Abstract

We propose two numerical methods for accelerating the convergence of the standard fixed point method associated with a nonlinear and/or degenerate elliptic partial differential equation. The first method is linearly stable, while the second is provably convergent in the viscosity solution sense. In practice, the methods converge at a nearly linear complexity in terms of the number of iterations required for convergence. The methods are easy to implement and do not require the construction or approximation of the Jacobian. Numerical examples are shown for Bellman’s equation, Isaacs’ equation, Pucci’s equations, the Monge–Ampère equation, a variant of the infinity Laplacian, and a system of nonlinear equations.

Appendix

We provide a proof of Proposition 4.2 below. The arguments follows from a direct computation of the lower bound.

Proof

We can expand $\alpha _n$ in terms of the sequences $\{\gamma _j\}^n_{j=1}$ as follows:

$$\begin{aligned} \alpha _n&=(1+\alpha _{n-1})\gamma _n=(1+(1+\alpha _{n-2})\gamma _{n-1})\gamma _n \\&=(1+(1+(1+\alpha _{n-2})\gamma _{n-2})\gamma _{n-1})\gamma _n\ldots \\&=\sum ^{n}_{j=1} \prod ^{n}_{k=j} \gamma _k \end{aligned}$$

Next, define $\xi _k{:}{=}1-\gamma _k$, by the assumptions of Proposition 4.2, $1-\xi _k \ge 1- \frac{1}{k+1}$, so

$$\begin{aligned} \prod ^{n}_{k=j} (1-\xi _k) \ge \prod ^{n}_{k=j} (1 - \frac{1}{k+1}) =\frac{j}{n+1}. \end{aligned}$$

Therefore the sequence $\alpha _n$ can be bound below by:

$$\begin{aligned} \alpha _n&=\sum ^{n}_{j=1} \prod ^{n}_{k=j} \gamma _k \ge \sum ^{n}_{j=1} \frac{j}{n+1} = \frac{n}{2}. \end{aligned}$$

Also, the partial sums of $\alpha _n$ are given by:

$$\begin{aligned} \sum ^{n-1}_{j=1} \alpha _j \ge \sum ^{n-1}_{j=1} \frac{j}{2} = \frac{n^2-n}{4}. \end{aligned}$$