Incremental unknowns for solving partial differential equations
Numerische Mathematik
Staggered incremental unknowns for solving Stokes and generalized Stokes problems
Applied Numerical Mathematics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
On a class of preconditioners for solving the Helmholtz equation
Applied Numerical Mathematics
A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems
SIAM Journal on Scientific Computing
Journal of Scientific Computing
Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian
SIAM Journal on Scientific Computing
Applied Numerical Mathematics
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A multilevel preconditioner for the Helmholtz equation using two types of incremental unknowns (IU) is developed. The transition between the two occurs when the mesh size reaches a predetermined fraction of the wavelength - roughly one quarter wavelength. Conventional IUs based on bilinear interpolation are employed for fine meshes like in an earlier paper by the authors. For coarse mesh sizes, novel IUs are defined using a Helmholtz/wave equation-based interpolation. The interpolation coefficients for the coarse meshes with dimension higher than one are derived numerically for stencils resembling integral representations for interior points. In two dimensions, the IUs are located on crosses surrounded by square contours. Numerical experiments on 1D and 2D examples have demonstrated the efficacy of the proposed approach in reducing the condition numbers and accelerating convergence for coarse grids with mesh sizes exceeding the wavelength.