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Scripted foilset CPS615 Finite Element and Conjugate Gradient Presentation

Given by Geoffrey C. Fox at CPS615 Fall Semester 95 Simulation Track on 18 November 95. Foils prepared 8 Nov 1995
Outside Index Summary of Material

This derives the finite element method for a simple two dimensional Laplacian with triangular elements
We use this to motivate conjugate gradient as a variant of steepest descent for variational principle underlying FEM
We discuss preconditioning, parallelism and convergence of general conjugate gradient method

Table of Contents for full HTML of CPS615 Finite Element and Conjugate Gradient Presentation

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1 CPS615 -- Base Course for the Simulation Track of Computational Science
Fall Semester 1995 --
Finite Element Methods and Conjugate Gradient Methods
2 Abstract of CPS615 Finite Element/Conjugate Gradient Presentation
3 19:Integral Formulation of Finite Element Method
4 20:Variation in Integral
5 21:Equivalence of Integral and Differential Formulation of Laplace's Equation
6 22:Discretization of Integral
7 23:Triangular Elements in Two Dimensions
8 24:Example for Two-Dimensional Triangular Elements
9 25:Bilinear Form of Integral with Triangular Elements
10 26:Formula for Stiffness Matrix Element
11 27:Finite Element Equations
12 28:Structure of Stiffness Matrix and Its Assembly
13 29:Conditions on Triangulation
14 30:Introduction to Poor Person's Conjugate Gradient
15 31:Conjugate Gradient Iteration for Quadratic Form
16 32:Conjugate Gradient and Method of Steepest Descent
17 33:Conjugate Gradient for Finite Element Problems
18 34:Poor Person's Conjugate Gradient and Eigenvalues of Matrix
19 35:Diagonalization of Quadratic Form
20 36:Diagonalization of Conjugate Gradient Equations
21 37:Convergence of Conjugate Gradient in Diagonalized Form
22 38:Clarification of Eigenvalue Analysis for Conjugate Gradient and Jacobi Iteration
23 39:Intuitive Description of Poor Person's Conjugate Gradient Algorithm
24 40:Improvement of Poor Person's Conjugate Gradient with Orthonormal Iteration
25 41:Full Conjugate Gradient Algorithm
26 42:Overview of Parallelism in Conjugate Gradient
27 43:Parallel Issues in Calculation of Matrix Elements
28 44:Scalar Products in Parallel Conjugate Gradient
29 45:Preconditioning in Conjugate Gradient
30 46:Convergence of Conjugate Gradient

Outside Index Summary of Material

HTML version of Scripted Foils prepared 8 Nov 1995

Foil 1 CPS615 -- Base Course for the Simulation Track of Computational Science
Fall Semester 1995 --
Finite Element Methods and Conjugate Gradient Methods

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
Geoffrey Fox
NPAC
Room 3-131 CST
111 College Place
Syracuse NY 13244-4100
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 2 Abstract of CPS615 Finite Element/Conjugate Gradient Presentation

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
This derives the finite element method for a simple two dimensional Laplacian with triangular elements
We use this to motivate conjugate gradient as a variant of steepest descent for variational principle underlying FEM
We discuss preconditioning, parallelism and convergence of general conjugate gradient method
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 3 19:Integral Formulation of Finite Element Method

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 4 20:Variation in Integral

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 5 21:Equivalence of Integral and Differential Formulation of Laplace's Equation

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 6 22:Discretization of Integral

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 7 23:Triangular Elements in Two Dimensions

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 8 24:Example for Two-Dimensional Triangular Elements

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 9 25:Bilinear Form of Integral with Triangular Elements

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 10 26:Formula for Stiffness Matrix Element

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 11 27:Finite Element Equations

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 12 28:Structure of Stiffness Matrix and Its Assembly

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 13 29:Conditions on Triangulation

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 14 30:Introduction to Poor Person's Conjugate Gradient

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 15 31:Conjugate Gradient Iteration for Quadratic Form

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 16 32:Conjugate Gradient and Method of Steepest Descent

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 17 33:Conjugate Gradient for Finite Element Problems

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 18 34:Poor Person's Conjugate Gradient and Eigenvalues of Matrix

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 19 35:Diagonalization of Quadratic Form

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 20 36:Diagonalization of Conjugate Gradient Equations

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 21 37:Convergence of Conjugate Gradient in Diagonalized Form

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 22 38:Clarification of Eigenvalue Analysis for Conjugate Gradient and Jacobi Iteration

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 23 39:Intuitive Description of Poor Person's Conjugate Gradient Algorithm

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 24 40:Improvement of Poor Person's Conjugate Gradient with Orthonormal Iteration

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 25 41:Full Conjugate Gradient Algorithm

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 26 42:Overview of Parallelism in Conjugate Gradient

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 27 43:Parallel Issues in Calculation of Matrix Elements

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 28 44:Scalar Products in Parallel Conjugate Gradient

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 29 45:Preconditioning in Conjugate Gradient

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
HTML version of Scripted Foils prepared 8 Nov 1995

Foil 30 46:Convergence of Conjugate Gradient

From CPS615 Finite Element and Conjugate Gradient Presentation CPS615 Fall Semester 95 Simulation Track -- 18 November 95. *
Full HTML Index
See Original Foil
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