SOLVING ELLIPTIC PDES

Transcription

1 university-logo SOLVING ELLIPTIC PDES School of Mathematics Semester

2 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

3 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

4 POISSON S EQUATION Equation and Boundary Conditions Solving the Model Problem A prototype elliptic pde is Poisson s equation given by 2 φ x φ = f (x, y), y2 where f (x, y) is a known/given function The equation has to be solved in a domain D

5 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

6 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.

7 THE MODEL PROBLEM Equation and Boundary Conditions Solving the Model Problem Consider the problem 2 φ x 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.

8 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.

9 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

10 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.

11 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

12 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

13 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.

14 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.

15 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.