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Finite Difference Discretization of Elliptic Equations: 1D Problem Lectures 2 and 3

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Model Problem Poisson 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 Differences Discretization Subdivide interval into equal subintervals for

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Numerical Solution Finite Differences Approximation For example … forsmall

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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 Isnon-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 Solution Finite 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 Analysis Properties of A -1 Let Non-negativity for Boundedness for

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Discretization Error Analysis Qualitative Properties of If for Then for

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Discretization Error Analysis Qualitative Properties of Discrete Stability

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Discretization Error Analysis Truncation Error For any we can show that Take

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Discretization Error Analysis Error Equation Let be the discretization error Subtracting and

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Discretization Error Analysis Error Equation

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Discretization Error Analysis Convergence Using the discrete stability estimate on or A-priori Error Estimate

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Discretization Error Analysis Numerical Example

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EXAMPLE : Asymptotically,

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Discretization Error Analysis Summary 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 the differential 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 errorsatisfies

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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 ConvergenceStability 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 Application Axially 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: Claim that Notesince

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The Eigenvalue Problem Error Analysis Analytical Solution:

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What are ?

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The Eigenvalue Problem Error Analysis Analytical Solution: 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 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 … Recall : or

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The Eigenvalue Problem Link to … Discretization … Error equation : for as(consistency)

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The Eigenvalue Problem Link to Norm Discretization We will use the modified norm for Thus, from consistency

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The Eigenvalue Problem Link to ||.|| Discretization … Ingredients: 1. Rayleigh Quotient : 2. Cauchy-Schwarz Inequality : for all

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The Eigenvalue Problem Link to ||.|| Discretization … Convergence proof:

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

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Alternative Derivation Since From error equation Multiplying by

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