Analysis and numerical simulation of the three-dimensional Cauchy problem for quasi-linear elliptic equations

This work is concerned with solving the Cauchy problem for quasilinear elliptic equations whose exponential instability is manifestly seen by the catastrophic growth in the representation of the exact solution. Our proposed regularization procedure consists in damping the unbounded terms in the representation. Moreover, we show that its solution converges to the exact solution uniformly and strongly in L² under a priori assumptions on the exact solution. In order to verify our analysis and the accuracy of the numerical procedures, we exhibit two numerical examples. Our main tools for simulation are the trigonometric polynomial approximation, and the fast Fourier transform in combination with the cubic Hermite interpolation.