Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics

By Barry Lee. Mathematics of Computation, 2006

1,273 citations recorded by the source corpus.

Subfields: Computational Mechanics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/9db9bf4f7ced835e9eb91e147cc6951d8cc74634

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Citations in the corpus

Some Preconditioning Techniques for Saddle Point Problems (2008, background): “Various constructions based on the Chebyshev polynomials can give more explicit convergence bounds, but these are somewhat less straightforward to write down for indefinite rather than definite symmetric matrices (see for example [23] or [13]).”
Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations (2015, background): “/q; for further details, see, for example, [46], Chapter 5.”
Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow (2007, background): “All Stokes preconditioners in IFISS have the structure proposed by Wathen and Silvester [1993] and Silvester and Wathen [1994], namely,”
Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data (2016, positive): “Thus, following [50], we proceed to derive the so-called Cahouet-Chabard approximation to S0 using a technique for the steady Navier-Stokes equation, which is based on the least squares commutator (see Chapter 8 of [14]) defined by E := (L)∇−∇(Lp), where L = (τ−1 + η)I + ∆ +K0 and Lp = (τ−1 + η)Ip +”
Primal‐dual active set methods for Allen–Cahn variational inequalities with nonlocal constraints (2010, background): “3) is in saddle point form which is known to arise in a variety of applications [14] and has been studied in detail for example in [14, 51].”
Stabilization of local projection type applied to convection-diffusion problems with mixed boundary conditions. (2008, background): “The presented examples, except the first one, can be found in [7, 26].”
hp analysis of a hybrid DG method for Stokes flow (2013, methodology): “For illustration of the theoretical results, we consider two test problems from [28].”
Solvers for large-displacement fluid–structure interaction problems: segregated versus monolithic approaches (2008, methodology): “The current implementation of this preconditioner within oomph- lib uses Elman, Silvester & Wathen’s “least squares commutator” (LSC) preconditioner [9] to approximately solve the fluid system (6).”
On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime (2014, background): “In the last forty years mixed finite elements for the incompressible NavierStokes equations have seen a great success in mathematical fluid dynamics [30, 8, 43, 18, 33, 16].”
High Accuracy Mantle Convection Simulation through Modern Numerical Methods. II: Realistic Models and Problems (2017, methodology): “…the form A B T B + C 0 , we use the preconditioner proposed by Silvester and Wathen for the symmetric Stokes system (see [Silvester & Wathen(1994), Elman et al.(2005)] for a derivation): where a tilde indicates an approximation and S = B T A − 1 B is the Schur Speciﬁcally, our test problem…”
Chebyshev semi-iteration in preconditioning for problems including the mass matrix. (2008, methodology): “…matrix vector products with A and compute a sequence of iterates {x k } from a starting guess x 0 which belong to nested Krylov subspaces x k ∈ x 0 + span{r 0 , Ar 0 , A 2 r 0 , . . . , A k−1 r 0 } for k = 1, 2, . . . , n, where r k = b − Ax k is the residual; see, for example, [8] [14] [26].”
A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations (2011, methodology): “All test problems were generated under Matlab using the IFISS software package [16] (see also [18]).”
FAST ITERATIVE SOLVERS FOR CONVECTION-DIFFUSION CONTROL PROBLEMS ∗ (2013, methodology): “The method discussed in [7] for solving (2.”