AERO4630-Project 1 Solved

Problem 1: Method of manufactured solutions
                                                                                       −∇2u(x) = f (x),         x ∈ Ω

                                                                                               u(x) = uD(x),         x ∈∂Ω

In the class, we did the example where

2              2 uD = 1 + x1 + 2x2,         f (x) = 6.0

The solution to the Laplace equation in this case is

2                2 u(x) = 1 + x1 + 2x2

We can verify this by computing the laplacian of u(x) as
(1.1)

(1.2)

(1.3)

(1.4)
∂2u ∂2u

 + = 2 + 2 × 2 = 6 = −f (x) ∂x2 ∂x2
(1.5)
solution using your code. This is called theequationA common way to verify that your code is working is to start with a known solution and then try to obtain thatmethod of manufactured solutions. Let’s look at the original Poisson

Let’s try this out for more cases.                       1                   2

(1a) Edit the python script and show results generated by Paraview. Report the errors.Try uknown(x) = uD(x) = x13 and f (x) = −6x1. First show that this satisfies the Poisson equation. Next,

(1b)                  Repeat the above with uknown(x) = uD(x) = x23 and f (x) = −6x2.

Problem 2: Improving accuracy Let’s go back to the above problem and take

uknown(x) = uD(x) = x13,                                                                                         f (x) = −6x1                                                                                                     (2.1)

You must have noticed error of the order to ∼ 0.001 or more. Let’s try to improve this.

(2a)           One way of improving the error is to make your mesh higher resolution. Try increasing the mesh fromelements:to  . Has the error reduced? Plot the L2 error norm and max error for increasing number ofand . Include the plots in your submission.

                  8 × 8        16 × 16

(2b) Another way to improve is to change the shape function. So far we have been dealing with LagrangePolynomial of ordermight not do a good job approximating it. Increase the order of the polynomial. Plot the L2 error norm4,8,16,32 . These are basically linear functions. If the solution64                                                         and These simulations will take time... be patient.is cubic, the linear function

1                                                                                                                                                            uknown

 and max error for increasing order: 1,2,3,4