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Factorizing the factorization – a spectral-element solver for elliptic equations with linear operation count. (English) Zbl 1380.65373
Summary: The paper proposes a novel factorization technique for static condensation of a spectral-element discretization matrix that yields a linear operation count of just $$13N$$ multiplications for the residual evaluation, where $$N$$ is the total number of unknowns. In comparison to previous work, it saves a factor larger than 3 and outpaces unfactored variants for all polynomial degrees. Using the new technique as a building block for a preconditioned conjugate gradient method yields linear scaling of the runtime with $$N$$ which is demonstrated for polynomial degrees from 2 to 32. This makes the spectral-element method cost effective even for low polynomial degrees. Moreover, the dependence of the iterative solution on the element aspect ratio is addressed, showing only a slight increase in the number of iterations for aspect ratios up to 128. Hence, the solver is very robust for practical applications.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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