99 research outputs found
A p-multigrid method enhanced with an ILUT smoother and its comparison to h-multigrid methods within Isogeometric Analysis
Over the years, Isogeometric Analysis has shown to be a successful
alternative to the Finite Element Method (FEM). However, solving the resulting
linear systems of equations efficiently remains a challenging task. In this
paper, we consider a p-multigrid method, in which coarsening is applied in the
approximation order p instead of the mesh width h. Since the use of classical
smoothers (e.g. Gauss-Seidel) results in a p-multigrid method with
deteriorating performance for higher values of p, the use of an ILUT smoother
is investigated. Numerical results and a spectral analysis indicate that the
resulting p-multigrid method exhibits convergence rates independent of h and p.
In particular, we compare both coarsening strategies (e.g. coarsening in h or
p) adopting both smoothers for a variety of two and threedimensional
benchmarks
Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids
In this paper, we first construct a nonconforming finite element pair for the
incompressible Stokes problem on quadrilateral grids, and then construct a
discrete Stokes complex associated with that finite element pair. The finite
element spaces involved consist of piecewise polynomials only, and the
divergence-free condition is imposed in a primal formulation. Combined with
some existing results, these constructions can be generated onto grids that
consist of both triangular and quadrilateral cells
Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion
We present a class of spline finite element methods for time-domain wave
propagation which are particularly amenable to explicit time-stepping. The
proposed methods utilize a discontinuous Galerkin discretization to enforce
continuity of the solution field across geometric patches in a multi-patch
setting, which yields a mass matrix with convenient block diagonal structure.
Over each patch, we show how to accurately and efficiently invert mass matrices
in the presence of curved geometries by using a weight-adjusted approximation
of the mass matrix inverse. This approximation restores a tensor product
structure while retaining provable high order accuracy and semi-discrete energy
stability. We also estimate the maximum stable timestep for spline-based finite
elements and show that the use of spline spaces result in less stringent CFL
restrictions than equivalent piecewise continuous or discontinuous finite
element spaces. Finally, we explore the use of optimal knot vectors based on L2
n-widths. We show how the use of optimal knot vectors can improve both
approximation properties and the maximum stable timestep, and present a simple
heuristic method for approximating optimal knot positions. Numerical
experiments confirm the accuracy and stability of the proposed methods
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