university-logo SOLVING ELLIPTIC PDES School of Mathematics Semester 1 2008
OUTLINE 1 REVIEW 2 POISSON S EQUATION Equation and Boundary Conditions Solving the Model Problem 3 THE LINEAR ALGEBRA PROBLEM Expressing the problem as a matrix equation The Linear System 4 SUMMARY
university-logo DIFFERENCING We need to know that a scheme is stable for it to be convergent Use stencils to generate difference formula for derivatives Change the centering of the approximation: to increase stability or the order of the scheme See notes for more differencing formulas
POISSON S EQUATION Equation and Boundary Conditions Solving the Model Problem A prototype elliptic pde is Poisson s equation given by 2 φ x 2 + 2 φ = f (x, y), y2 where f (x, y) is a known/given function The equation has to be solved in a domain D
DOMAIN OF THE PROBLEM Equation and Boundary Conditions Solving the Model Problem Boundary conditions are given on the boundary D of D. y dd D x
BOUNDARY CONDITIONS Equation and Boundary Conditions Solving the Model Problem These can be of three types: Dirichlet φ = g(x, y) on D. = g(x, y) on D. ( ) Robin/Mixed B = 0 on D. Neumann φ n φ, φ n Robin boundary conditions involve a linear combination of φ and its normal derivative on the boundary. Mixed boundary conditions involve different conditions for one part of the boundary, and another type for other parts of the boundary.
THE MODEL PROBLEM Equation and Boundary Conditions Solving the Model Problem Consider the problem 2 φ x 2 + 2 φ = f (x, y), 0 x, y 1 y2 with boundary conditions φ = 0 on D. Here the domain D is the square region 0 < x < 1 and 0 < y < 1.
DISCRETISING THE PROBLEM Equation and Boundary Conditions Solving the Model Problem Construct a finite difference mesh with points (x i, y j ), where x i = i x, for i = 0, 1,...,N, and y j = j y, for j = 0, 1,...M. Here x = 1/N, and y = 1/M are the step sizes in the x and y directions.
DISCRETISING THE PROBLEM Equation and Boundary Conditions Solving the Model Problem Next replace the derivatives in Poisson equation by the discrete approximations to get: w i+1,j 2w i,j + w i 1,j x 2 + w i,j+1 2w i,j + w i,j 1 y 2 = f i,j, for 1 i N 1, and 1 j M 1. At the boundary we have the equations w i,j = 0, if i = 0, N and 0 j M w i,j = 0, if j = 1, M and 0 i N The result is a system of (N 1) (M 1) equations with (N 1) (M 1) unknowns, The unknowns are w i,j in the region D. university-logo
THE SOLUTION AS A VECTOR Expressing the problem as a matrix equation The Linear System Let us write the solution w i,j as w i = (w i,1, w i,2,...,w i,m 1 ) T and also the right hand side of the equation as f i = (f i,1, f i,2,...,f i,m 1 ) T Then we can use this notation to write the problem in matrix form.
THE MATRIX EQUATION Expressing the problem as a matrix equation The Linear System B I I B I I B I I B w 1 w 2 w 3 w N 1 = x 2 f 1 f 2 f 3 f N 1
Expressing the problem as a matrix equation The Linear System In the above I is the (M 1) (M 1) identity matrix The matrix B is given by B = b c a b c a b c...... where a = c = β 2, and b = 2(1 + β 2 ) β = x y a b
THE LINEAR SYSTEM Expressing the problem as a matrix equation The Linear System Then let us write the linear system as Aw = f We observe that the matrix A is very sparse. The matrix A is very large.
SOLVING THE LINEAR SYSTEM Expressing the problem as a matrix equation The Linear System In the next lecture we look into how to solve the problem. Methods can generally be split into: Direct Methods: Expensive, requiring large storage. Iterative Methods: Method of choice in most cases.
university-logo We have shown how to discretise a model elliptic problem. The discretised equations can be expressed as the matrix equation. Aw = f Help us think about methods in an abstract way. The matrix form will allow us to analyse the stability and convergence of schemes.