Hiptmair–Xu preconditioner

In mathematics, Hiptmair–Xu (HX) preconditioners[1] are preconditioners for solving

H(curl){displaystyle H(operatorname {curl} )}

and

H(div){displaystyle H(operatorname {div} )}

problems based on the auxiliary space preconditioning framework.[2] An important ingredient in the derivation of HX preconditioners in two and three dimensions is the so-called regular decomposition, which decomposes a Sobolev space function into a component of higher regularity and a scalar or vector potential. The key to the success of HX preconditioners is the discrete version of this decomposition, which is also known as HX decomposition. The discrete decomposition decomposes a discrete Sobolev space function into a discrete component of higher regularity, a discrete scale or vector potential, and a high-frequency component.

HX preconditioners have been used for accelerating a wide variety of solution techniques, thanks to their highly scalable parallel implementations, and are known as AMS[3] and ADS[4] precondition. HX preconditioner was identified by the U.S. Department of Energy as one of the top ten breakthroughs in computational science[5] in recent years. Researchers from Sandia, Los Alamos, and Lawrence Livermore National Labs use this algorithm for modeling fusion with magnetohydrodynamic equations.[6] Moreover, this approach will also be instrumental in developing optimal iterative methods in structural mechanics, electrodynamics, and modeling of complex flows.

. . . Hiptmair–Xu preconditioner . . .

Consider the following

H(curl){displaystyle H(operatorname {curl} )}

problem: Find

uHh(curl){displaystyle uin H_{h}(operatorname {curl} )}

such that

(curl u,curl v)+τ(u,v)=(f,v),vHh(curl),{displaystyle (operatorname {curl} ~u,operatorname {curl} ~v)+tau (u,v)=(f,v),quad forall vin H_{h}(operatorname {curl} ),}

with

τ>0{displaystyle tau >0}

.

The corresponding matrix form is

Acurlu=f.{displaystyle A_{operatorname {curl} }u=f.}

The HX preconditioner for

H(curl){displaystyle H(operatorname {curl} )}

problem is defined as

Bcurl=Scurl+ΠhcurlAvgrad1(Πhcurl)T+gradAgrad1(grad)T,{displaystyle B_{operatorname {curl} }=S_{operatorname {curl} }+Pi _{h}^{operatorname {curl} },A_{vgrad}^{-1},(Pi _{h}^{operatorname {curl} })^{T}+operatorname {grad} ,A_{operatorname {grad} }^{-1},(operatorname {grad} )^{T},}

where

Scurl{displaystyle S_{operatorname {curl} }}

is a smoother (e.g., Jacobi smoother, Gauss–Seidel smoother),

Πhcurl{displaystyle Pi _{h}^{operatorname {curl} }}

is the canonical interpolation operator for

Hh(curl){displaystyle H_{h}(operatorname {curl} )}

space,

Avgrad{displaystyle A_{vgrad}}

is the matrix representation of discrete vector Laplacian defined on

[Hh(grad)]n{displaystyle [H_{h}(operatorname {grad} )]^{n}}

,

grad{displaystyle grad}

is the discrete gradient operator, and

Agrad{displaystyle A_{operatorname {grad} }}

is the matrix representation of the discrete scalar Laplacian defined on

Hh(grad){displaystyle H_{h}(operatorname {grad} )}

. Based on auxiliary space preconditioning framework, one can show that

κ(BcurlAcurl)C,{displaystyle kappa (B_{operatorname {curl} }A_{operatorname {curl} })leq C,}

where

κ(A){displaystyle kappa (A)}

denotes the condition number of matrix

A{displaystyle A}

.

In practice, inverting

Avgrad{displaystyle A_{vgrad}}

and

Agrad{displaystyle A_{grad}}

might be expensive, especially for large scale problems. Therefore, we can replace their inversion by spectrally equivalent approximations,

Bvgrad{displaystyle B_{vgrad}}

and

Bgrad{displaystyle B_{operatorname {grad} }}

, respectively. And the HX preconditioner for

H(curl){displaystyle H(operatorname {curl} )}

becomes

Bcurl=Scurl+ΠhcurlBvgrad(Πhcurl)T+gradBgrad(grad)T.{displaystyle B_{operatorname {curl} }=S_{operatorname {curl} }+Pi _{h}^{operatorname {curl} },B_{vgrad},(Pi _{h}^{operatorname {curl} })^{T}+operatorname {grad} B_{operatorname {grad} }(operatorname {grad} )^{T}.}

. . . Hiptmair–Xu preconditioner . . .

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. . . Hiptmair–Xu preconditioner . . .

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