Finite Difference Discretization of Elliptic Equations: 1D Problem Lectures 2 and 3.
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Transcript of Finite Difference Discretization of Elliptic Equations: 1D Problem Lectures 2 and 3.
![Page 1: Finite Difference Discretization of Elliptic Equations: 1D Problem Lectures 2 and 3.](https://reader035.fdocuments.in/reader035/viewer/2022081505/551502c2550346a80c8b5499/html5/thumbnails/1.jpg)
Finite Difference Discretizationof Elliptic Equations: 1D Problem
Lectures 2 and 3
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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
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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
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Numerical Finite DifferencesDiscretization
Subdivide interval into equal subintervals
for
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Numerical Solution Finite Differences Approximation
For example …
for small
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Numerical Solution Finite Differences Equations
suggests …
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Numerical Solution Finite Differences ….Equations
(Symmetric)
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Numerical Solution Finite Differences Solution
Is non-singular ?
For any
Hence for any ( Is APD)
exists and is unique
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Numerical Solution Finite Differences Example…
with
Take
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Numerical SolutionFinite Differences …Example
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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 ?
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Discretization Error AnalysisProperties of A-1
Let
• Non-negativity
for
• Boundedness
for
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Discretization Error AnalysisQualitative Properties of u
0ˆ 0 uf
If
for
Then
for
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Discretization Error Analysis Qualitative Properties of
Discrete Stabilityu
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Discretization Error AnalysisTruncation Error
For any we can show that
Take
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Discretization Error AnalysisError Equation
Let be the discretization error
Subtracting
and
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Discretization Error AnalysisError Equation
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Discretization Error AnalysisConvergence
Using the discrete stability estimate on
or
A-priori Error Estimate
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Discretization Error AnalysisNumerical Example
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Discretization Error AnalysisNumerical Example
EXAMPLE :
Asymptotically,
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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 .
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Generalizations Definitions
Consider a linear elliptic differential equation
and a difference scheme
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Generalizations Consistency
The difference scheme is consistent with thedifferential equation if:
For all smooth functions
for
when
for all
is order of accuracy
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Generalizations Truncation Error
or,
for
The truncation error results from inserting the exact solution into the difference scheme.
Consistency
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Generalizations Error Equation
Original scheme
Consistercy
The error satisfies
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Generalizations Stability
Matrix norm
The difference scheme is stable if
(independent of )
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Generalizations Stability
(max row sum)
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Generalizations Convergence
Error equation
Taking norms
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Generalizations Summary
Consistency + Stability Convergence
Convergence Stability Consistency
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The Eigenvalue Problem Model Problem Statement
Find nontrivial such that
denote solutions with
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The Eigenvalue Problem ApplicationAxially Loaded Beam
• “Small” Deflection
• External Force
Equilibrium
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The Eigenvalue Problem Exact Solution
or
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The Eigenvalue Problem Exact Solution
Thus (choose )
Larger more oscillatory larger
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The Eigenvalue Problem Exact Solution
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The Eigenvalue Problem Discrete Equations Difference
Formulas
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The Eigenvalue Problem Discrete Equations Matrix Form
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The Eigenvalue Problem Error Analysis
Analytical Solution: ...ˆ,ˆ kku
Claim that
Note since
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The Eigenvalue Problem Error Analysis
Analytical Solution: ...ˆ,ˆ kku
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The Eigenvalue Problem Error Analysis
Analytical Solution: ...ˆ,ˆ kku
What are ?
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The Eigenvalue Problem Error Analysis
Analytical Solution: ...ˆ,ˆ kku Thus:
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The Eigenvalue Problem Error Analysis Conclusions…
Low modes
For fixed , :
second-order convergence
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The Eigenvalue Problem Error Analysis …Conclusions…
High modes :
For
as
High modes ( ) are not accurate.
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The Eigenvalue Problem Error Analysis …Conclusions…
Low modes vs. high modes
Example :
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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.
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The Eigenvalue Problem Error Analysis …Conclusions…
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The Eigenvalue Problem Error Analysis …Conclusions…
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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.
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The Eigenvalue Problem Link to …Discretization…fuxx
Recall :
or
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The Eigenvalue Problem Link to …Discretization…fuxx
Error equation :
for
as (consistency)
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The Eigenvalue Problem Link to Norm Discretizationfuxx
We will use the “modified” norm
for
Thus, from consistency
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The Eigenvalue Problem Link to ||.|| Discretization…fuxx
Ingredients:
1. Rayleigh Quotient :
2. Cauchy-Schwarz Inequality :
for all
for all
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The Eigenvalue Problem Link to ||.|| Discretization…fuxx
Convergence proof:
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The Eigenvalue Problem Link to ||.|| Discretization…fuxx
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The Eigenvalue Problem Link to ||.|| Discretization…fuxx
Alternative DerivationSince
From error equation
Multiplying by