IN5270 -  Numerical methods for partial differential equations - Solved

Problem 1 Solving a ID stationary linear PDE with finite elements (weight 50%)

The following ID PDE

                                 —uxx. + au f G), O < < L                            (1)

is to be solved, where is a given function and a > O is a scalar constant. Neumann boundary conditions apply on both the boundary points, more specifically, we have ux = C at = O and ux D at L.

la (weight 6%)
Please derive in detail the weak variational form of (1) for the purpose of using the Galerkin method.

1b (weight 10%)
We will use a non-uniform mesh with in total Ne elements, where all elements are of the PI type. Suppose element number e is of length he, such that EN_0 1 he = L. Explain in detail how to compute, with help of a standardized reference PI element, the element matrix associated with an interior element, that is, with element index O < e < Ne 1. (There is no need to compute the element vector.)

lc (weight 10%)
Explain in detail how the element matrix will be, respectively, for the leftmost element (with index e O) and the rightmost element (with index e = Ne — 1). Please also present the formulas for computing the element vectors for these two elements. (But you don't need to compute the actual values in these element vectors.)

Id (weight 12%)
How many rows will there be in the global matrix that arises from assembling all the Ne element matrices? What is the total number of nonzero values in the global matrix? Explain in detail what the actual values will be on row number i in the global matrix.

le (weight 12%)
If the boundary condition on the left point O is changed to a socalled Robin condition:           (2)

3.)

where r and B are scalar constants. (The boundary condition on the right boundary point remains unchanged.)

Please explain in detail how the weak variational form will change due to the above change of boundary condition. Please also explain the exact changes that will happen to the resulting global matrix and global vector.

Problem 2 Solving a 2D nonlinear diffusion equation with finite differences (weight 50%)

We consider the following 2D nonlinear diffusion equation:

  + f(u), @,y) e Q = (O, 1) 2 , t e (0, T], (3)

where K > O is a scalar constant and f (u) is a given nonlinear function in u. The initial condition is of the form

 

and the homogeneous Neumann boundary condition, O, is applicable on the entire boundary (that is, the four sides =  x = 1 and y = 1).

2a (weight 5%)
Use the backward Euler scheme in time and derive the resulting time discrete problem for each time step, with At as the time step size.

2b (weight 12%)
Suppose we use a 2D uniform mesh in space with I'Vx small intervals in the direction and Ny small intervals in the y direction. The values of I'VX and Ny may be different. If central finite differences are used to discretize the time discrete problem from question 2a, what will be the resulting system of nonlinear algebraic equations? (How does each nonlinear algebraic equation look like? How many nonlinear algebraic equations are there in total?)

2c (weight 8%)
Explain in detail how Picard iterations can be used to solve the system of nonlinear algebraic equations from question 2b.

4.)

2d (weight 15%)
Explain in detail how Newton's method can be used to solve the system of nonlinear algebraic equations from question 2b.

2e (weight 10%)
Suppose you have implemented a computer program, based on the backward Euler scheme in time, central finite differencing in space, and Picard iterations or Newton's method as above, to solve the 2D nonlinear diffusion equation (3). This code can freely change the mesh resolution Ax, Ay, At.

If the code is to be run by your friend on a powerful computer and there is a I-hour time limit, how would you recommend the choice of Ax, Ay and At with the purpose of obtaining a highest possible accuracy of the numerical solution?

Remarks: Your friend can tell you beforehand the chosen value of T, but not the actual choices of 1@, y) and f(u). You can experiment with the code beforehand as much as possible on your own PC, but you don't have access to your friend's powerful computer. The powerful computer has S times the speed of your own PC, and your friend has just "one go" of a I-hour execution on the powerful computer.