A geometric optics ansatz-based plane wave method for two dimensional Helmholtz equations with variable wave numbers

In this paper we develop a plane wave type method for discretization of homogeneous Helmholtz equations with variable wave numbers. In the proposed method, local basis functions (on each element) are constructed by the geometric optics ansatz such that they approximately satisfy a homogeneous Helmholtz equation without boundary condition. More precisely, each basis function is expressed as the product of an exponential plane wave function and a polynomial function, where the phase function in the exponential function approximately satisfies the eikonal equation and the polynomial factor is recursively determined by transport equations associated with the considered Helmholtz equation. We prove that the resulting plane wave spaces have high order $h$-approximations as the standard plane wave spaces (which are available only to the case with constant wave number). We apply the proposed plane wave spaces to the discretization of nonhomogeneous Helmholtz equations with variable wave numbers and establish the corresponding error estimates of their finite element solutions. We report some numerical results to illustrate the efficiency of the proposed method

A diagonal sweeping domain decomposition method with source transfer for the Helmholtz equation

In this paper, we propose and test a novel diagonal sweeping domain decomposition method (DDM) with source transfer for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$. In the method the computational domain is partitioned into overlapping checkerboard subdomains for source transfer with the perfectly matched layer (PML) technique, then a set of diagonal sweeps over the subdomains are specially designed to solve the system efficiently. The method improves the additive overlapping DDM (W. Leng and L. Ju, 2019) and the L-sweeps method (M. Taus, et al., 2019) by employing a more efficient subdomain solving order. We show that the method achieves the exact solution of the global PML problem with $2^n$ sweeps in the constant medium case. Although the sweeping usually implies sequential subdomain solves, the number of sequential steps required for each sweep in the method is only proportional to the $n$-th root of the number of subdomains when the domain decomposition is quasi-uniform with respect to all directions, thus it is very suitable for parallel computing of the Helmholtz problem with multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to demonstrate the effectiveness and efficiency of the proposed method.Comment: 39 pages, 17 figure

A novel least squares method for Helmholtz equations with large wave numbers

In this paper we are concerned with numerical methods for nonhomogeneous Helmholtz equations in inhomogeneous media. We design a least squares method for discretization of the considered Helmholtz equations. In this method, an auxiliary unknown is introduced on the common interface of any two neighboring elements and a quadratic subject functional is defined by the jumps of the traces of the solutions of local Helmholtz equations across all the common interfaces, where the local Helmholtz equations are defined on elements and are imposed Robin-type boundary conditions given by the auxiliary unknowns. A minimization problem with the subject functional is proposed to determine the auxiliary unknowns. The resulting discrete system of the auxiliary unknowns is Hermitian positive definite and so it can be solved by the PCG method. Under some assumptions we show that the generated approximate solutions possess almost the optimal error estimates with little "wave number pollution". Moreover, we construct a substructuring preconditioner for the discrete system of the auxiliary unknowns. Numerical experiments show that the proposed methods are very effective for the tested Helmholtz equations with large wave numbers.Comment: 34 page

L-Sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation

We present the first fast solver for the high-frequency Helmholtz equation that scales optimally in parallel, for a single right-hand side. The L-sweeps approach achieves this scalability by departing from the usual propagation pattern, in which information flows in a 180 degree cone from interfaces in a layered decomposition. Instead, with L-sweeps, information propagates in 90 degree cones induced by a checkerboard domain decomposition (CDD). We extend the notion of accurate transmission conditions to CDDs and introduce a new sweeping strategy to efficiently track the wave fronts as they propagate through the CDD. The new approach decouples the subdomains at each wave front, so that they can be processed in parallel, resulting in better parallel scalability than previously demonstrated in the literature. The method has an overall O((N/p) log w) empirical run-time for N=n^d total degrees-of-freedom in a d-dimensional problem, frequency w, and p=O(n) processors. We introduce the algorithm and provide a complexity analysis for our parallel implementation of the solver. We corroborate all claims in several two- and three-dimensional numerical examples involving constant, smooth, and discontinuous wave speeds