Finite Difference Discretization of Elliptic Equations FD Formulas

Finite Difference Discretization of Elliptic Equations: FD Formulas and Multidimensional Prblems Lecture 4

Finite Difference Formulas Problem Definition Given distinct points , find the weights that is of optimal order of accuracy. Two approaches : • Lagrange interpolation • undetermined coefficients such

Finite Difference Formulas Langrage interpolation Lagrange polynomials Lagrange interpolant

Finite Difference Formulas Langrage interpolation Approximate Therefore,

Finite Difference Formulas Langrage interpolation Example… Set Second order Lagrange interpolant

Finite Difference Formulas Langrage interpolation …Example… Assuming a uniform grid (First derivative) Forward Centered Backward

Finite Difference Formulas Langrage interpolation …Example (Second derivative) Centered

Finite Difference Formulas Undetermined coefficients Start from Insert Taylor expansions for Determine coefficients about to maximize accuracy.

Finite Difference Formulas Undetermined coefficients Example… (uniform spacing )

Finite Difference Formulas Undetermined coefficients …Example Equating coefficients of Solve,

Poisson Equation in 2 D Definition in on

Poisson Equation in 2 D Discretization

Poisson Equation in 2 D Approximation For example … for small

Poisson Equation in 2 D Equations suggests ….

Poisson Equation in 2 D Equations Example…

Poisson Equation in 2 D Equations …Example

Poisson Equation in 2 D Equations Numbering becomes component

Poisson Equation in 2 D Equations Block Matrix… Block tridiagonal matrix

Poisson Equation in 2 D Equations …Block Matrix Block Definitions has a banded structure Bandwidth :

Poisson Equation in 2 D Equations SPD Property Hence for any : ( is SPD) exists and is unique

Poisson Equation in 2 D Error Analysis Truncation Error For for all

Poisson Equation in 2 D Error Analysis Stability It can be shown that Ingredients: • Positivity of the coeffiicients of • Bound on the maximum row sum

Poisson Equation in 2 D Error Analysis Stability Error equation If

Eigenvalue Problem in 2 D Statement in on Assume (for simplicity) Solutions

Eigenvalue Problem in 2 D Exact Solution Eigenvalues Eigenvectors

Eigenvalue Problem in 2 D Discrete Problem Eigenvectors

Eigenvalue Problem in 2 D Discrete Problem Eigenvalues Low Modes High Modes ( , fixed) as

Eigenvalue Problem in 2 D Condition Number of A as If grows (in ) as number of grid points. (better than in 1 D, relatively speaking !!)

Eigenvalue Problem in 2 D Link to Error Estimate Error equation

Discrete Fourier Solution Poisson Problem 2 D is SPD diagonal matrix of eigenvalues is matrix of eigenvectors

Discrete Fourier Solution Poisson Problem 2 D ALGORITHM Still cost is … But …

Discrete Fourier Solution Poisson Problem 2 D • Matrix multiplications can be reorganized (tensor product evaluation) • is a (Inverse) Discrete Fourier Transform Using FFT

Discrete Fourier Solution Poisson Problem 2 D We are interested in solving in on on where and are given.

Non-Rectangular Domains Poisson Problem 2 D Mapping Can we determine an equivalent problem to be solved on ?

Non-Rectangular Domains Poisson Problem 2 D Transformed equations How can we evaluate terms and ?

Non-Rectangular Domains Poisson Problem 2 D …Transformed equations…

Non-Rectangular Domains Poisson Problem 2 D …Transformed equations… and

Non-Rectangular Domains Poisson Problem 2 D …Transformed equations Finally becomes and depend on the mapping

Non-Rectangular Domains Poisson Problem 2 D Normal Derivatives… is parallel to (or ); e. g. , on

Non-Rectangular Domains Poisson Problem 2 D Normal Derivatives… Thus, with