where is the Laplace operator and is a scalar function.

Transcription

1 Elliptic PDEs A brief discussion of two important elliptic PDEs. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as: 2 where is the Laplace operator and is a scalar function. 2 f Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson. (Wikipedia)

2 An elliptic PDE models the steady state behavior. For example, the potential energy of a point in a plane acted on by gravitational forces in the plane, two-dimensional steady-state problems involving incompressible fluids (A fluid which is not reduced in volume by an increase in pressure.), and the steady state distribution of heat on a plane region whose boundary is being held at specific temperatures. Suppose that u(x, y) is the temperature at point (x, y) in a planar region. Then Poisson s equation is given by the expression Here we will assume that Poisson s equation is specified on a rectangular domain with u(x, y) = g(x, y) on the boundary S of region R. The Laplacian operator It is also denoted by the symbol.

3 Poisson s equation Note that from the general form of our linear PDE A = 1, B = 0 and C = 1 so B 2 4AC < 0, hence the equation is elliptic. We impose a grid on the domain R as shown.

4 Poisson s equation We use finite difference approximations for the partial derivatives. To denote the fact that we indeed have approximations will be replaced by. Here we have Using these approximations we have local truncation error O(h 2 + k 2 ) and the resulting approximation to Poisson s equation is given by the difference expression for i = 1, 2,, n-1 and j = 1, 2,, m-1. (The interior grid/mesh points.)

5 At the interior grid points. From the boundary condition u(x, y) = g(x, y) we have We can rearrange EQN #1 into the following form by multiplying both sides by h 2 and collecting like terms to obtain This approximation scheme is used to generate estimates of u at interior grid points of the rectangular domain.

6 Inspection of the grid points used in EQN#2 shows that to estimate by w ij we use information from compass point neighbors. This diagram represents the stencil for Poisson s equation.

7 Special case: If h = k then in EQN#2 the coefficients are simplified so we have the following simpler form for EQN#2

8 The general system of equations represented by for i = 1, 2,, n-1 and j = 1, 2,, m-1 is linear in the unknowns so it can be written in matrix form. In order to efficiently work with the square coefficient matrix which has size (n 1)(m 1) we need a numbering scheme for the unknowns. Different numbering schemes can result in various structures in the coefficient matrix.

9 A common numbering scheme is lexicographic ordering. We will start at the bottom left corner of the computational grid and number the unknowns from left to right and from bottom to top. For example suppose the rectangular domain has n = 4 and m = 3, then the 5 by 4 grid (it is 5 by 4 because we must account for the boundary) is shown in the figure below where the number written next to the location of an interior grid point is the lexicographic number associated with the unknown. (The grid is 5 by 4 to accommodate the boundary conditions. The indexing of grid points goes from 0 to n and 0 to m. Also note that the interior has (n- 1)(m-1) grid points.)

10 Let s assume that h = k, then we can use the simpler expressions given by If we place the stencil over the unknown #1 then i = 1 and j = 1 so we have However, are values from the boundary conditions; Thus the first equation becomes If move the stencil over the unknown #2 then i = 2 and j = 1 so we have However, is a value from the boundary conditions; Thus the second equation becomes

11 By moving the stencil to the unknowns in lexicographic ordering the resulting linear system will be 6 6 for the grid shown and the coefficient matrix will be To see the structure of the coefficient matrix let and I 3 be the 3 3 identity matrix. Then we have If we had h = k and 9 unknowns then the coefficient matrix would be of the form where the 0 represents an appropriately size block of zeros. It can be shown that for any choices of h and k the coefficient matrix is nonsingular and block tridiagonal, hence the linear system has a unique solution.

12 Example: Consider the Poisson equation On the domain with boundary conditions For the grid we use 6 interior points where n = 4 and m = 3. The approximations at the interior points in lexicographic order can be shown to be

13

14 With n = 5 and m = 5 there are16 interior points and the solution in lexicographic order is

15 The solution to this elliptic PDE is so we can compute the error involved. Here we consider the root mean square (rms) error which is defined as In the case of n = 4, m = 3 we obtain rms error e-004 while in the case of n = 5, m = 5 the rms error e-004.

16 Using MATLAB: We have a poisson program that must be edited to set function f(x, y) and the boundary conditions for each problem and then saved under a new name. In this way we preserve format of the program for the example given above. The program used for the example is poisson_ex_bradie_9_1(0,1,0,1/2,4,3) for the 6 interior points poisson_ex_bradie_9_1(0,1,0,1/2,5,5) for the 16 interior points The parameter list is xl, xr, yb, yt, n, m where the interval for x is [xl, xr], the interval for y is [yb, yt], n is the number of equispaced subintervals in the x-direction and m is the number of equispaced subintervals in the y-direction. (Note there will be (n - 1)(m 1) interior points.

17 The output will be a list of the (n - 1)(m 1) approximations in lexicographical order starting at the bottom of the rectangular domain and listed from left to right. This will be display as 2-dimensional array whose columns are approximations at each value of y in the order of the values of x. For poisson_ex_bradie_9_1(0,1,0,1/2,4,3) the output is which are in the order In addition, a three dimensional plot of the solution will be displayed. You can rotate this figure using the rotation icon in the MATLAB toolbar. To edit the program to insert function and boundary conditions at a MATLAB prompt type edit poisson_ex_bradie_9_1. Once the editor displays the code scroll down until you see the following code lines.

18 To edit the program to insert function and boundary conditions at a MATLAB prompt type edit poisson_ex_bradie_9_1. Once the editor displays the code scroll down until you see the following code lines. The function must be in terms of x and y. Each of expressions for the boundaries must be coded using operators.*,./, and.^ in order to get the dimensions correct. Once you have changed the assignments for f and the boundary conditions save the file under a different name to the MATLAB folder containing my library. (I suggest a name like poisson_problem_99 or an appropriate number.) To execute the modified program at a MATLAB prompt enter poisson_problem_99(xl,xr,yb,yt,n,m) with values from the problem replacing the names. If you use poisson_problem_99(xl,xr,yb,yt,n,m); note the semicolon you only get the figure