finite difference discretization of elliptic equations: 1d problem lectures 2 and 3

Finite Difference Discretizationof Elliptic Equations: 1D Problem

Lectures 2 and 3

Model ProblemPoisson Equation in 1D

Boundary Value Problem (BVP)

Describes many simple physical phenomena (e.g.):• Deformation of an elastic bar• Deformation of a string under tension• Temperature distribution in a bar

Model Problem Poisson Equation in 1D Solution Properties

• The solution always exists

• is always “smoother” then the data

• If for all , then for all

•

• Given the solution is unique

Numerical Finite DifferencesDiscretization

Subdivide interval into equal subintervals

for

Numerical Solution Finite Differences Approximation

For example …

for small

Numerical Solution Finite Differences Equations

suggests …

Numerical Solution Finite Differences ….Equations

(Symmetric)

Numerical Solution Finite Differences Solution

Is non-singular ?

For any

Hence for any ( Is APD)

exists and is unique

Numerical Solution Finite Differences Example…

with

Take

Numerical SolutionFinite Differences …Example

Numerical Solution Finite Differences Convergence ?

1. Does the discrete solution retain the qualitative propeties of the continuous solution ?

2. Does the solution become more accurate when ?

3. Can we make for arbierarily small ?

Discretization Error AnalysisProperties of A-1

Let

• Non-negativity

for

• Boundedness

for

Discretization Error AnalysisQualitative Properties of u

0ˆ 0 uf

If

for

Then

for

Discretization Error Analysis Qualitative Properties of

Discrete Stabilityu

Discretization Error AnalysisTruncation Error

For any we can show that

Take

Discretization Error AnalysisError Equation

Let be the discretization error

Subtracting

and

Discretization Error AnalysisError Equation

Discretization Error AnalysisConvergence

Using the discrete stability estimate on

or

A-priori Error Estimate

Discretization Error AnalysisNumerical Example

Discretization Error AnalysisNumerical Example

EXAMPLE :

Asymptotically,

Discretization Error AnalysisSummary

• For a simple model problem we can produce numerical approximations of arbitrary accuracy.

• An a-priori error estimate gives the asymptotic dependence of the solution error on the discretization size .

Generalizations Definitions

Consider a linear elliptic differential equation

and a difference scheme

Generalizations Consistency

The difference scheme is consistent with thedifferential equation if:

For all smooth functions

for

when

for all

is order of accuracy

Generalizations Truncation Error

or,

for

The truncation error results from inserting the exact solution into the difference scheme.

Consistency

Generalizations Error Equation

Original scheme

Consistercy

The error satisfies

Generalizations Stability

Matrix norm

The difference scheme is stable if

(independent of )

Generalizations Stability

(max row sum)

Generalizations Convergence

Error equation

Taking norms

Generalizations Summary

Consistency + Stability Convergence

Convergence Stability Consistency

The Eigenvalue Problem Model Problem Statement

Find nontrivial such that

denote solutions with

The Eigenvalue Problem ApplicationAxially Loaded Beam

• “Small” Deflection

• External Force

Equilibrium

The Eigenvalue Problem Exact Solution

or

The Eigenvalue Problem Exact Solution

Thus (choose )

Larger more oscillatory larger

The Eigenvalue Problem Exact Solution

The Eigenvalue Problem Discrete Equations Difference

Formulas

The Eigenvalue Problem Discrete Equations Matrix Form

The Eigenvalue Problem Error Analysis

Analytical Solution: ...ˆ,ˆ kku

Claim that

Note since

The Eigenvalue Problem Error Analysis

Analytical Solution: ...ˆ,ˆ kku

The Eigenvalue Problem Error Analysis

Analytical Solution: ...ˆ,ˆ kku

What are ?

The Eigenvalue Problem Error Analysis

Analytical Solution: ...ˆ,ˆ kku Thus:

The Eigenvalue Problem Error Analysis Conclusions…

Low modes

For fixed , :

second-order convergence

The Eigenvalue Problem Error Analysis …Conclusions…

High modes :

For

as

High modes ( ) are not accurate.

The Eigenvalue Problem Error Analysis …Conclusions…

Low modes vs. high modes

Example :

The Eigenvalue Problem Error Analysis …Conclusions…

Low modes vs. high modes

resolved

accurate

not resolved

not accurate

is

BUT: as , , so any fixed mode converges.

The Eigenvalue Problem Error Analysis …Conclusions…

The Eigenvalue Problem Error Analysis …Conclusions…

The Eigenvalue Problem Condition Number of A

For a SPD matrix , the condition number is

given by

=maximum eigenvalue of

minimum eigenvalue of

Thus, for our matrix,

as

grows (in ) as number of grid points squared.

Importance: understanding solution procedures.

The Eigenvalue Problem Link to …Discretization…fuxx

Recall :

or

The Eigenvalue Problem Link to …Discretization…fuxx

Error equation :

for

as (consistency)

The Eigenvalue Problem Link to Norm Discretizationfuxx

We will use the “modified” norm

for

Thus, from consistency

The Eigenvalue Problem Link to ||.|| Discretization…fuxx

Ingredients:

1. Rayleigh Quotient :

2. Cauchy-Schwarz Inequality :

for all

for all

The Eigenvalue Problem Link to ||.|| Discretization…fuxx

Convergence proof:

The Eigenvalue Problem Link to ||.|| Discretization…fuxx

The Eigenvalue Problem Link to ||.|| Discretization…fuxx

Alternative DerivationSince

From error equation

Multiplying by