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Erschienen in:
High Order Whitney Forms on Simplices and the Question of Potentials Kapitel lesen Erstes Kapitel lesen

2021 | OriginalPaper | Buchkapitel

Assembly of Multiscale Linear PDE Operators

verfasst von : Miroslav Kuchta

Erschienen in: Numerical Mathematics and Advanced Applications ENUMATH 2019

Verlag: Springer International Publishing

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Abstract

In numerous applications the mathematical model consists of different processes coupled across a lower dimensional manifold. Due to the multiscale coupling, finite element discretization of such models presents a challenge. Assuming that only singlescale finite element forms can be assembled we present here a simple algorithm for representing multiscale models as linear operators suitable for Krylov methods. Flexibility of the approach is demonstrated by numerical examples with coupling across dimensionality gap 1 and 2. Preconditioners for several of the problems are discussed.

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Vorheriges Kapitel On Mesh Regularity Conditions for Simplicial Finite Elements
Nächstes Kapitel A Least-Squares Galerkin Gradient Recovery Method for Fully Nonlinear Elliptic Equations
Fußnoten
1
Implementation can be found in the Python module FEniCS iihttps://​github.​com/​MiroK/​fenics_​ii.
 
2
The serial performance of our pure Python implementation is cca. 2× slower than the native FEniCS implementation [12]. More precisely, assembling (1) on \(\Omega =\left [0, 1\right ]^2\) discretized by 2 ⋅ 10242 triangles and continuous linear Lagrange elements (the system matrix size is cca. 106, however, it is not explicitly formed here) takes 3.86 s (to be compared with 1.79 s). Most of the time is spent building T h. The trace matrix is reused by the interpreter to evaluate both B h and \(B^{\prime }_h\).
 
3
Details of experimental setup. We discretize Ωi uniformly by first dividing the domains into n × m rectangles and afterwords splitting each rectangle into two triangles. For Ω1 we have m = n, m = 2n for Ω2 so that the trace meshes of the domains are different. Krylov solvers are started from random initial guess. Convergence tolerance for relative preconditioned residual norm of 10−10 is used. Unless specified otherwise the preconditioner blocks use LU factorization.
 
4
Finite element space of continuous Lagrange elements of order k is denoted by P k while RT 0 denotes the space of lowest order Raviart-Thomas elements.
 
5
The implementation of the Darcy-Stokes problems with conforming meshes can be found at https://​github.​com/​MiroK/​fenics_​ii/​blob/​master/​demo/​ as dq_darcy_stokes_2d.py (primal formulation) and darcy_stokes_2d.py (mixed formulation).
 
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Metadaten
Titel
Assembly of Multiscale Linear PDE Operators
verfasst von
Miroslav Kuchta
Copyright-Jahr
2021
Buch
Numerical Mathematics and Advanced Applications ENUMATH 2019

Print ISBN: 978-3-030-55873-4

Electronic ISBN: 978-3-030-55874-1

Copyright-Jahr: 2021

DOI
https://doi.org/10.1007/978-3-030-55874-1_63