The Fast Adaptive Composite Grid (FAC) Method for Elliptic Equations

McCormick, Steve; Thomas, J. W.

doi:10.2307/2007986

Cited by 72 publications

(36 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To solve the discretized linear system on the composite mesh

S_{ℓ}

Find u_{ℓ} \in V_{ℓ} such that a_{ℓ} false(u_{ℓ}, v_{ℓ} false) = {⟨ g_{ℓ}, v_{ℓ} ⟩}_{ℓ} \forall v_{ℓ} \in V_{ℓ},

we use the FAC methodology . This is based on the overlapping composite‐grid decomposition of

S_{ℓ}

, where each “patch”

T_{ℓ, k}

serves as a level in the multigrid hierarchy.…”

Section: Fac Methodsmentioning

confidence: 99%

“…The FAC methodology was first proposed in the 1980s to extend optimal multigrid performance to meshes generated by structured adaptive mesh refinement algorithms. Many such algorithms generate meshes that are semistructured , which can be characterized by having distinct “levels” of refinement with regular structure within each level.…”

Section: Introductionmentioning

confidence: 99%

“…FAC, in contrast to AMG, explicitly requires that the composite grid consist of distinct mesh levels, each of which can be considered independently of the others. FAC multigrid cycles consist of independent processing on each level, coupled by intergrid‐transfer operators as in standard multigrid, and can be done both multiplicatively or additively (asynchronously) . We note that many other generalizations of the multigrid methodology are possible, including popular subspace and domain decomposition methods; however, the hierarchical structure of FAC maps naturally onto the hierarchical structure of local mesh refinement that will be considered here.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Composite‐grid multigrid for diffusion on the sphere

Adler

Lashuk

MacLachlan

2017

Numerical Linear Algebra App

View full text Add to dashboard Cite

SummaryRecently, there has been much interest in the solution of differential equations on surfaces and manifolds, driven by many applications whose dynamics take place on such domains. Although increasingly powerful algorithms have been developed in this field, many straightforward questions remain, particularly in the area of coupling advanced discretizations with efficient linear solvers. In this paper, we develop a structured refinement algorithm for octahedral triangulations of the surface of the sphere. We explain the composite-grid finite-element discretization of the Laplace-Beltrami operator on such triangulations and extend the fast adaptive composite-grid scheme to provide an efficient solution of the resulting linear system. Supporting numerical examples are presented, including the recovery of second-order accuracy in the case of a nonsmooth solution. KEYWORDSfast adaptive composite-grid (FAC) algorithm, finite-element discretizations on surfaces, Laplace-Beltrami operator, multigrid

show abstract

“…To solve the discretized linear system on the composite mesh

S_{ℓ}

Find u_{ℓ} \in V_{ℓ} such that a_{ℓ} false(u_{ℓ}, v_{ℓ} false) = {⟨ g_{ℓ}, v_{ℓ} ⟩}_{ℓ} \forall v_{ℓ} \in V_{ℓ},

we use the FAC methodology . This is based on the overlapping composite‐grid decomposition of

S_{ℓ}

, where each “patch”

T_{ℓ, k}

serves as a level in the multigrid hierarchy.…”

Section: Fac Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Composite‐grid multigrid for diffusion on the sphere

Adler

Lashuk

MacLachlan

2017

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…Then as the finer levels are coarsened they are absorbed, if possible, into coarser levels. This method of relaxation on an adaptive hierarchy is in contrast to the Fast Adaptive Composite Grid Method [65], though we do not believe that the particular details of how the elliptic equations are solved have much bearing on the algorithm described here for dealing with coupled elliptic/hyperbolic equations.…”

Section: Multigrid On An Adaptive Hierarchymentioning

confidence: 99%

Adaptive mesh refinement for coupled elliptic-hyperbolic systems

Pretorius¹,

Choptuik²

2006

Journal of Computational Physics

View full text Add to dashboard Cite

We present a modification to the Berger and Oliger adaptive mesh refinement algorithm designed to solve systems of coupled, non-linear, hyperbolic and elliptic partial differential equations. Such systems typically arise during constrained evolution of the field equations of general relativity. The novel aspect of this algorithm is a technique of "extrapolation and delayed solution" used to deal with the non-local nature of the solution of the elliptic equations, driven by dynamical sources, within the usual Berger and Oliger time-stepping framework. We show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations, and further demonstrate that the solution time scales approximately linearly with problem size. We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress spurious highfrequency solution components ("noise").

show abstract

“…It is desirable that the computational effort of the error estimation process be only a fraction of the finite element solution, hence in practice only an approximation for p is calculated. We will be seeking for an approximation of by replacing the Hessian matrix u(4,4) resulting from (35) by its diagonal. The resulting energy norm of the estimated error for each element is obtained by where NDOFS are the total number of degrees of freedom in the element 'e'.…”

mentioning

confidence: 99%

THEs-VERSION OF FINITE ELEMENT METHOD FOR LAMINATED COMPOSITES

Fish

Guttal

1996

Int. J. Numer. Meth. Engng.

View full text Add to dashboard Cite

The s-version of the finite element method is developed for laminated plates and shells. By this technique the global domain is idealized using 2-D Equivalent Single Layer model. The regions where ESL model errs badly in capturing localized phenomena are superimposed by a stack of 3-D elements. Assumed strain formulation and selective polynomial order escalation in the two models, as well as fast iterative procedures, are employed to maintain a high level of computational efficiency.

show abstract

The Fast Adaptive Composite Grid (FAC) Method for Elliptic Equations

Cited by 72 publications

References 11 publications

Composite‐grid multigrid for diffusion on the sphere

Composite‐grid multigrid for diffusion on the sphere

Adaptive mesh refinement for coupled elliptic-hyperbolic systems

THEs-VERSION OF FINITE ELEMENT METHOD FOR LAMINATED COMPOSITES

Contact Info

Product

Resources

About