PAMM · Proc. Appl. Math. Mech. 17, 129 – 132 (2017) / DOI 10.1002/pamm.201710037
Building blocks for a leading edge high-order ﬂow solver
, Jörg Stiller
, and Jochen Fröhlich
Institute of Fluid Mechanics, TU Dresden
Center for Advancing Electronics Dresden (cfaed)
An important trend in Computational Fluid Dynamics is towards high-order methods, as they offer a substantially lower
discretization error for the same number of degrees of freedom (DOF). Examples are the Spectral-Element Methods (SEM)
and Discontinuous Galerkin (DG) methods. Unfortunately, with most implementations the work load of such solvers increases
drastically with the number of DOF, for example, when increasing the polynomial degree of the approximation. This issue
gets particular pressing for elliptic solvers which are a vital building block in the time-stepping of the incompressible Navier-
Stokes equations, resulting from pressure projection methods or implicit treatment of viscous terms. So far, this drastic
increase of resources has hampered the use of SEM for higher polynomial degrees, such as 16 or more.
The present contribution is located at this particular “Frontier of CFD” and proposes an SEM with linear scaling in the
number of degrees of freedom. It is achieved independent of the polynomial degree, independent of the aspect ratio of
elements, and involves constant iteration count when increasing the number of elements. The method is based on combining
static condensation with block-Jacobi preconditioning and iterative substructuring. The latter two lead to a constant iteration
count, while the former warrants linear operator complexity.
In the presentation, the scheme is described in detail and applied to the construction of a Helmholtz solver. This solver is
extremely fast and able to solve the Helmholtz equation with 10
unknowns on 240 cores in acceptable time. It thus enables
competitive high-order SEM simulations even on small clusters and in this way expands the frontier of CFD towards highly
accurate results requiring comparatively modest resources.
2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Current research in Computational Fluid Dynamics focuses on high-order methods, such as the Discontinuous Galerkin (DG)
and Spectral-Element Method (SEM). They allow for higher convergence rates at the expense of more complex operators.
However, fast elliptic solvers are still a matter of research for these methods.
For a perfect solver, the number of iterations should be independent of the number of degrees of freedom, i.e. for the SEM
the polynomial degree and the number of elements, as well as robust against the increase of the aspect ratio of the elements.
This allows for a linearly scaling runtime, should the operation count of the operator and preconditioner scale linearly as well.
Current methods do not hold up to this standard. Multigrid with overlapping block-smoothers [1, 2] is a good candidate, as
it generates a constant iteration count, but it is not robust against the aspect ratio and the operators scale super-linearly when
increasing the polynomial degree. The ﬁrst problem can be remedied by introducing weighting functions, which works for
SEM  as well as DG , however the scaling still remains. Techniques combining iterative substructuring [5, 6] and static
condensation , as done in , create robustness against increases of the polynomial degree and a constant iteration count.
Yet, the static condensed operator scales super-linearly. Another candidate is the fast static condensation, e.g. [9, 10], which
allows for a linearly scaling operator and robustness against increases of the polynomial degree. But the technique does not
include a global coupling. It, hence, is not robust against increases of the number of elements, and while combining it with
the Cascadic Multigrid Method can lead to an iteration count independent of the number of elements, it depends on the target
solution , rendering it less effective.
This paper proposes a solver that combines fast static condensation with iterative substructuring, leading to a solver that
scales linearly when increasing the number of elements or the polynomial degree. The resulting condition number is indepen-
dent of the number of elements and has only a slight increase with the polynomial degree.
2 Static condensation in the spectral-element method
The HELMHOLTZ equation, sometimes referred to as elliptic equation, λu − ∆u = f in a domain Ω, with a solution variable u,
continuous right-hand side f, parameter λ ≥ 0, and ∆ as the Laplacian, is a main ingredient of many solvers for incompress-
ible ﬂuid ﬂow. Introducing the spectral-element method as discretization, see e.g. [12, 13], results in a discrete algebraic
system Hu = F with H being the discrete HELMHOLTZ operator, u as solution variable, and F as the discrete right-hand
side. For the SEM, the operator works on the global degrees of freedom, but is typically evaluated on an element-by-element
Corresponding author: e-mail email@example.com
2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim