From: Zhuoran Wang Date: Thu, 19 Dec 2019 17:33:49 +0000 (-0700) Subject: Add initial version of results.dox X-Git-Tag: v9.2.0-rc1~678^2~15 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=5d8969227161c305c53bcd7d8cd22d3e804b05ee;p=dealii.git Add initial version of results.dox --- diff --git a/examples/step-71/doc/results.dox b/examples/step-71/doc/results.dox index b5eaba9377..13ba2a879c 100644 --- a/examples/step-71/doc/results.dox +++ b/examples/step-71/doc/results.dox @@ -1,2 +1,155 @@

Results

+We run the program with a right hand side that will produce the +solution $u = \sin(\pi x) \sin(\pi y)$ and with clamped +boundary conditions in the domain $\Omega = (0,1)^2$. +We test this setup using $Q_2$, $Q_3$, and $Q_4$ elements, which one can +change `fe\_degree` in `main()`. With mesh +refinement, the $L_2$ convergence rates, $H_1$-seminorm convergence +and $H_2$-seminorm convergence of $u$ +should then be around 2, 2, 1 for $Q_2$ , 4, 3, 2 for +$Q_3$, and 5, 4, 3 for $Q_4$ separately. +We use different penalties $\eta = 1$, $2$, and $p(p+1)$ where $p$ +is the degree of polynomials, +and compare convergence rates of numerical solutions computed by these +penalties. + + +

Test results on Q2 with \eta = p(p+1)

+ +

Convergence table

+ +We run the code with differently refined meshes +and get the following convergence rates. + + + + + + + + + + + + + + + + + +
number of refinements $\|u-u_h^\circ\|_{L_2}$ Conv. rates $|u-u_h|_{H_1}$ Conv. rates $|u-u_h|_{H_2}$ Conv. rates
2 1.539e-02 8.528e-02 1.602
3 4.563e-03 1.75 2.408e-02 1.82 7.965e-01 1.00
4 1.250e-03 1.86 6.438e-03 1.90 3.969e-01 1.00
5 3.277e-04 1.93 1.666e-03 1.94 1.981e-01 1.00
+We can see that the $L_2$ convergence rates are around 2, +$H_1$-seminorm convergence rates are around 2, +and $H_2$-seminorm convergence rates are around 1. + +

Test results on Q3 with \eta = p(p+1)

+ +

Convergence table

+ + + + + + + + + + + + + + + + + + +
number of refinements $\|u-u_h^\circ\|_{L_2}$ Conv. rates $|u-u_h|_{H_1}$ Conv. rates $|u-u_h|_{H_2}$ Conv. rates
2 2.187e-04 4.46269e-03 1.638e-01
3 1.334e-05 4.03 5.54622e-04 3.00 4.095e-02 2.00
4 8.273e-07 4.01 6.90599e-05 3.00 1.023e-02 2.00
5 5.164e-08 4.00 8.62168e-06 3.00 2.558e-03 2.00
+We can see that the $L_2$ convergence rates are around 4, +$H_1$-seminorm convergence rates are around 3, +and $H_2$-seminorm convergence rates are around 2. +This, of course, matches our theoretical expectations. + +

Test results on Q4 with \eta = p(p+1)

+ +

Convergence table

+ + + + + + + + + + + + + + + + + + +
number of refinements $\|u-u_h^\circ\|_{L_2}$ Conv. rates $|u-u_h|_{H_1}$ Conv. rates $|u-u_h|_{H_2}$ Conv. rates
2 8.34446e-06 0.000239323 0.0109785
3 2.98497e-07 4.80 1.63221e-05 3.87 0.0013551 3.01
4 9.87063e-09 4.91 1.06066e-06 3.94 0.000167898 3.01
5 7.88939e-10 3.64 6.75478e-08 3.97 2.08912e-05 3.00
+We can see that the $L_2$ norm convergence rates are around 5, +$H_1$-seminorm convergence rates are around 4, +and $H_2$-seminorm convergence rates are around 3. +On the finest mesh, the $L_2$ norm convergence rate +is much smaller than our theoretical expectations +because the linear solver becomes the limiting factor due +to round-off. But the $L_2$ error is pretty small in that case. + +

Test results on Q2 with \eta = 1

+ +

Convergence table

+ + + + + + + + + + + + + + + + + + +
number of refinements $\|u-u_h^\circ\|_{L_2}$ Conv. rates $|u-u_h|_{H_1}$ Conv. rates $|u-u_h|_{H_2}$ Conv. rates
2 4.86048e-02 3.30386e-01 4.34917
3 1.29921e-02 1.90 1.4852e-01 1.15 4.01192 0.116
4 3.33539e-03 1.96 7.20252e-02 1.04 3.96138 0.018
5 8.41058e-04 1.98 3.57705e-02 1.00 3.95719 0.001
+Although $L_2$ norm and $H_1$-seminorm convergence rates of $u$ +follow the theoretical expectations, $H_2$-seminorm does not converge. +Comparing results from $\eta = 1$ and $\eta = p(p+1)$, +$\eta = p(p+1)$ is a better penalty. + +

Test results on Q2 with \eta = 2

+ +

Convergence table

+ + + + + + + + + + + + + + + + + + +
number of refinements $\|u-u_h^\circ\|_{L_2}$ Conv. rates $|u-u_h|_{H_1}$ Conv. rates $|u-u_h|_{H_2}$ Conv. rates
2 5.482e-03 7.652e-02 1.756e-00
3 2.227e-02 1.29 2.177e-02 1.81 8.711e-01 1.01
4 9.088e-04 1.29 6.026e-03 1.85 4.196e-01 1.05
5 2.822e-04 1.68 1.605e-03 1.90 2.041e-01 1.03
+In the case, all convergence rates of $u$ +follow the theoretical expectations. +But compared to the results from $\eta = p(p+1)$, +it does not show a good convergence on $L_2$ errors.