# A Fourier-Chebyshev Spectral Method for Cavitation Computation in Nonlinear Elasticity

## TL;DR

This paper introduces a Fourier-Chebyshev spectral method for accurately solving cavitation problems in nonlinear elasticity, with proven convergence and efficient algorithms demonstrated through numerical experiments.

## Contribution

The paper develops a novel spectral method combining Fourier and Chebyshev techniques for cavitation in nonlinear elasticity, including error analysis and convergence proof.

## Key findings

- The method achieves high accuracy in cavitation simulations.
- Numerical experiments confirm the efficiency and precision of the approach.
- The approach converges reliably for the nonlinear elasticity cavitation problem.

## Abstract

A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is obtained, and the convergence of the method is proved. An algorithm combined a gradient type method with a damped quasi-Newton method is applied to solve the discretized nonlinear equilibrium equations. Numerical experiments show that the Fourier-Chebyshev spectral method is efficient and capable of producing accurate numerical cavitation solutions.