# Hiptmair–Xu preconditioner

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

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

${displaystyle H(operatorname {div} )}$ problems based on the auxiliary space preconditioning framework. 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 and ADS precondition. HX preconditioner was identified by the U.S. Department of Energy as one of the top ten breakthroughs in computational science in recent years. Researchers from Sandia, Los Alamos, and Lawrence Livermore National Labs use this algorithm for modeling fusion with magnetohydrodynamic equations. 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

${displaystyle H(operatorname {curl} )}$ problem: Find

${displaystyle uin H_{h}(operatorname {curl} )}$ such that

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

${displaystyle tau >0}$ .

The corresponding matrix form is

${displaystyle A_{operatorname {curl} }u=f.}$ The HX preconditioner for

${displaystyle H(operatorname {curl} )}$ problem is defined as

${displaystyle B_{operatorname {curl} }=S_{operatorname {curl} }+Pi _{h}^{operatorname {curl} },A_{vgrad}^{-1},(Pi _{h}^{operatorname {curl} })^{T}+operatorname {grad} ,A_{operatorname {grad} }^{-1},(operatorname {grad} )^{T},}$ where

${displaystyle S_{operatorname {curl} }}$ is a smoother (e.g., Jacobi smoother, Gauss–Seidel smoother),

${displaystyle Pi _{h}^{operatorname {curl} }}$ is the canonical interpolation operator for

${displaystyle H_{h}(operatorname {curl} )}$ space,

${displaystyle A_{vgrad}}$ is the matrix representation of discrete vector Laplacian defined on

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

${displaystyle grad}$ is the discrete gradient operator, and

${displaystyle A_{operatorname {grad} }}$ is the matrix representation of the discrete scalar Laplacian defined on

${displaystyle H_{h}(operatorname {grad} )}$ . Based on auxiliary space preconditioning framework, one can show that

${displaystyle kappa (B_{operatorname {curl} }A_{operatorname {curl} })leq C,}$ where

${displaystyle kappa (A)}$ denotes the condition number of matrix

${displaystyle A}$ .

In practice, inverting

${displaystyle A_{vgrad}}$ and

${displaystyle A_{grad}}$ might be expensive, especially for large scale problems. Therefore, we can replace their inversion by spectrally equivalent approximations,

${displaystyle B_{vgrad}}$ and

${displaystyle B_{operatorname {grad} }}$ , respectively. And the HX preconditioner for

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

${displaystyle B_{operatorname {curl} }=S_{operatorname {curl} }+Pi _{h}^{operatorname {curl} },B_{vgrad},(Pi _{h}^{operatorname {curl} })^{T}+operatorname {grad} B_{operatorname {grad} }(operatorname {grad} )^{T}.}$