[cig-commits] commit: More writing

[Mercurial]

changeset:   80:6da349163758
tag:         tip
user:        Matthew G Knepley <knepley at gmail.com>
date:        Tue Oct 25 17:35:54 2011 -0500
files:       faultRup.tex
description:
More writing


diff -r 571296eee6c2 -r 6da349163758 faultRup.tex
--- a/faultRup.tex	Fri Oct 21 10:10:49 2011 -0500
+++ b/faultRup.tex	Tue Oct 25 17:35:54 2011 -0500
@@ -763,8 +763,6 @@ construct many variations of effective p
 construct many variations of effective parallel preconditioners for
 saddle point problems.
 
-\matt{Add a few sentences about algebraic multi-grid}.
-
 The PCFIELDSPLIT \cite{PETSc:manual} preconditioner in PETSc allows
 the user to define sets of unknowns which correspond to different
 fields in the physical problem. This scheme is flexible enough to
@@ -833,7 +831,7 @@ A suitable block diagonal approximation 
 \begin{equation}
   \tensor{P}^{-1} = \left( \begin{array}{cc}
       \tensor{K}^{-1} & 0 \\
-      0 & -(\tensor{L} \tensor{K}^{-1} ]tensor{L}^T)^{-1}
+      0 & -(\tensor{L} \tensor{K}^{-1} \tensor{L}^T)^{-1}
     \end{array} \right),
 \end{equation}
 which leads to
@@ -848,11 +846,18 @@ which leads to
   \end{array} \right).
 \end{equation}
 
-We focus on evaluating the lower portion of the preconditioning matrix
-associated with the Lagrange multipliers and rely on PETSc
-preconditioners to construct the upper portion. We approximate
+The elastic block $K$ can be further split into blocks associated with displacements along the $x$, $y$, and $z$
+axes. It is known that the vector Laplacian is spectrally equivalent to this operator~\cite{AskMarkAdams}, and each
+component is efficiently preconditioned by Algebraic Multigrid (AMG), such as ML~\cite{ML}. Algebraic Multigrid mimics
+the action of traditional geometric multgrid, but it generates coarse level operators and interpolation matrices using
+only the system matrix, treated as a weighted graph, rather than a separate description of the problem geometry, such as
+a mesh. We use PCFIELDSPLIT to split the elastic block and separately apply AMG to each component.
+
+We will now focus on evaluating the lower portion of the preconditioning matrix
+associated with the Lagrange multipliers since stock PETSc
+preconditioners can handle the upper portion. We approximate
 $\tensor{K}^{-1}$ with the inverse of the diagonal portion of
-$\tensor{K}$. $tensor{L}$ consists of integrating the products of
+$\tensor{K}$. $\tensor{L}$ consists of integrating the products of
 basis functions over the fault faces. Its structure depends on the
 quadrature scheme and the choice of basis functions. For conventional
 low order finite-elements and Gauss quadrature, $\tensor{L}$ contains
@@ -868,15 +873,19 @@ portion of the conditioning matrix,
   \tensor{P}_f = -\tensor{L_p} (\tensor{K}_{n+n+} + \tensor{K}_{n-n-}) \tensor{K}_p^{T}.
 \end{equation}
 
+\brad{Matt: I conjecture that collocation, because it makes $\tensor{L}$ block diagonal, is more tolerant of the
+  diagonal approximation for $\tensor{K}$.}
+
 For the upper portion of the preconditioning matrix associated with
 elasticity, we have found algrebraic multigrid preconditioners
 provide substantially faster convergence that the Additive Schwarz
 method. We combine the field split preconditioner with the algebraic multigrid
-preconditioner (CITE ML), such that we precondition the degrees of
-freedom for each global coordinate axis independently.
-
-\matt{Table of convergence for a fault example with ASM, block
-  tridiagonal, block tridiagonal+custom preconditioner, LSC}
+preconditioner, such that we precondition the degrees of
+freedom for each global coordinate axis independently. Table~\ref{tab:iterates}
+shows the number of iterates required to solve a problem with three faults to a
+relative tolerance of $10^{-8}$. It clearly shows the surperiority of our custom
+fault preconditioner. It also reveals that while is scales fairly well, the custom
+preconditioner does have a dependence on problem size.
 
 \begin{table}
 \centering
@@ -894,6 +903,7 @@ GMRES & FS/ML/Custom    &          37 & 
   solver tolerance is $10^{-8}$, and the preconditioners used were Additive Schwarz Method (ASM), FieldSplit (FS),
   Incomplete Cholesky (ICC), algebraic multigrid (ML), Schur complement, least-square commutator (LSC), and a custom
   preconditioner for the fault problem.}
+\label{tab:iterates}
 \end{table}
 
 \subsection{Dynamic Simulations}