In this paper, we achieve three primary objectives related to the rigorous computational analysis of nonlinear PDEs posed on complex geometries such as disks and cylinders. First, we introduce a validated Matrix Multiplication Transform (MMT) algorithm, analogous to the discrete Fourier transform, which offers a reliable framework for evaluating nonlinearities in spectral methods while effectively mitigating challenges associated with rounding errors. Second, we examine the Zernike polynomials, a spectral basis well-suited for problems on the disk, and highlight their essential properties. We further demonstrate how the MMT approach can be effectively employed to compute the product of truncated Zernike series, ensuring both accuracy and efficiency. Finally, we combine the MMT framework and Zernike series to construct computer-assisted proofs that establish the existence of solutions to two distinct nonlinear elliptic PDEs on the disk.

翻译：本文围绕复杂几何区域（如圆盘和圆柱）上非线性偏微分方程的严格计算分析，实现了三个主要目标。首先，我们提出了一种验证矩阵乘法变换算法，该算法类似于离散傅里叶变换，为谱方法中非线性项的评估提供了可靠框架，同时有效缓解了舍入误差带来的挑战。其次，我们研究了适用于圆盘问题的谱基——泽尼克多项式，并阐明了其基本性质。我们进一步论证了如何利用MMT方法有效计算截断泽尼克级数的乘积，确保计算过程的精确性与高效性。最后，我们将MMT框架与泽尼克级数相结合，构建了计算机辅助证明，从而严格证实了圆盘上两类非线性椭圆型偏微分方程解的存在性。