Presentation on theme: "Finite Difference Discretization of Elliptic Equations: 1D Problem Lectures 2 and 3."— Presentation transcript:
Finite Difference Discretization of Elliptic Equations: 1D Problem Lectures 2 and 3
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
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 Differences Discretization Subdivide interval into equal subintervals for
Numerical Solution Finite Differences Approximation For example … forsmall
Numerical Solution Finite Differences Solution Isnon-singular ? For any Hence for any ( Is APD) exists and is unique
Numerical Solution Finite Differences Example … with Take
Numerical Solution Finite 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 Analysis Properties of A -1 Let Non-negativity for Boundedness for
Discretization Error Analysis Qualitative Properties of If for Then for
Discretization Error Analysis Qualitative Properties of Discrete Stability
Discretization Error Analysis Truncation Error For any we can show that Take
Discretization Error Analysis Error Equation Let be the discretization error Subtracting and
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.
Generalizations Definitions Consider a linear elliptic differential equation and a difference scheme
Generalizations Consistency The difference scheme is consistent with the differential 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 errorsatisfies
Generalizations Stability Matrix norm The difference scheme is stable if (independent of )
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 … Recall : or
The Eigenvalue Problem Link to … Discretization … Error equation : for as(consistency)
The Eigenvalue Problem Link to Norm Discretization We will use the modified norm for Thus, from consistency
The Eigenvalue Problem Link to ||.|| Discretization … Ingredients: 1. Rayleigh Quotient : 2. Cauchy-Schwarz Inequality : for all
The Eigenvalue Problem Link to ||.|| Discretization … Convergence proof:
The Eigenvalue Problem Link to ||.|| Discretization …
Alternative Derivation Since From error equation Multiplying by
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