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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