We develop a sparse hierarchical $hp$-finite element method ($hp$-FEM) for the Helmholtz equation with rotationally invariant variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse $hp$-FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. We consider examples such as a high-frequency Helmholtz equation with radial discontinuities, the time-dependent Schr\"odinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm.

翻译：本文针对二维圆盘或圆环区域上具有旋转不变变系数亥姆霍兹方程，提出一种稀疏分层$hp$-有限元方法（$hp$-FEM）。网格由一个内部圆盘单元（若为圆环区域则省略）和多个同心圆环单元构成。该离散化方案保留了旋转不变算子（如拉普拉斯算子）的傅里叶模式解耦特性，体现为块对角质量矩阵和刚度矩阵。此外，矩阵的稀疏模式与离散阶数无关，并支持最优复杂度的分解。所提出的稀疏$hp$-FEM能够处理右端项和旋转不变亥姆霍兹系数中的径向不连续性。我们通过高频含径向不连续性亥姆霍兹方程、含时薛定谔方程以及基于交替方向隐式（ADI）算法实现准最优求解的三维圆柱域扩展等算例验证了方法的有效性。