Here, we develop a Petrov-Galerkin spectral method for the fractional Helmholtz equations (FHEs) of even order ν = s + k,s ∈ (k − 1,k) and k ∈ℕ. We define trial and test functions by related generalized Jacobi functions (GJFs). Moreover, we efficiently establish the well-posedness of problem and provide the rigorous priori error estimates. Furthermore, by auxiliary equation, we also obtain the super-approximation estimates. Notably, we propose a post-processing technique for the Petrov-Galerkin spectral method, and give the error estimates of corrected solution. In addition, we define a posteriori error estimators, and prove that they are asymptotically accurate. Finally, we demonstrate the sharpness of our error estimates by numerical experiments.

在这里，我们为偶数阶ν = s + k , s ∈ ( k − 1, k ) 和k ∈ ℕ的分数阶亥姆霍兹方程 (FHE) 开发了一种 Petrov-Galerkin 谱方法. 我们通过相关的广义雅可比函数 (GJF) 定义试验和测试函数。此外，我们有效地建立了问题的适定性并提供了严格的先验误差估计。此外，通过辅助方程，我们还获得了超近似估计。值得注意的是，我们为 Petrov-Galerkin 谱方法提出了一种后处理技术，并给出了修正解的误差估计。此外，我们定义了后验误差估计量，并证明它们是渐近准确的。最后，我们通过数值实验证明了我们的误差估计的锐度。