Objectives-

• To understand the general procedure of finite element formulation.

• To appreciate convergence and learn error estimation.

A) The 2n° order ODE

(d²u/dx²) + sinx = 0

with boundary conditions u(0) = 0 and u'(2Π) =1 has analytical solution

u(x) = sin x

The Galerkin weak form of this ODE is

₀∫^2Π[- (du/dx)(dδu/dx) + sin x δu(x)] dx δu(2Π) = 0,      ∀Su(x)

with essential boundary condition u(0) = 0.

1) Implement the weak form in COMSOL, and experiment on the effect of element size and type. You may control the mesh by selecting "User-controlled mesh" under meshing and specific the maximum element size. For element type, you may open "Discretization" under "Weak form PDC' (if not shown, toggle on the option by clicking on the "eye" icon on top-left of the "Model builder").

Test Lagrange elements of all possible orders.

2) Calculate the L2 error of all cases of your study

||e||_L2 = [∫7 (U_approx -u_exact)²dx]^1/2

and plot the error of each element type (order) in one curve as a function of the element size.
Use log-log plot to show the general trend. You may evaluate the integral by right click on "ResultskDerived ValuesilntegrationiLine Integration", select the line, and type in the expression (integrand).

3) Re-plot the L2 error against the total number of nodes in each case (copy the results out and plot it in other software, say excel). Which element is more efficient?

4) Plot the distribution of the error of the derivative, e(x) = (du/dx) - cos x, for linear and quadratic elements (by meshing the interval into 1 or 2 element). Where are the results of derivatives more accurate?

B) Now consider the ODE

(d²u/dx²) + (1/4)x^3/2= 0

with essential boundary conditions u(0) = 0 and u(1) = 1. Repeat steps (1)-(3) of the last problem for this equation. The exact solution for this equation is

u(x) = x^1/2