1 Introduction
2 VD approximation for parabolic bilinear OCP
3 Error estimates of intermediate variables
4 Convergence analysis
5 Superconvergence analysis
6 Numerical experiments
h
|
k
|
\(|\!|\!|u-u_{h}|\!|\!|\)
| Rate |
\(|\!|\!|R_{h}y-y_{h}|\!|\!|_{1}\)
| Rate |
\(|\!|\!|R_{h}p-p_{h}|\!|\!|_{1}\)
| Rate |
---|---|---|---|---|---|---|---|
\(\frac{1}{10}\)
|
\(\frac{1}{10}\)
| 1.46324e − 02 | – | 3.35880e − 02 | – | 7.31505e − 02 | – |
\(\frac{1}{20}\)
|
\(\frac{1}{40}\)
| 3.91388e − 03 | 1.90 | 8.18124e − 03 | 2.04 | 1.89129e − 02 | 1.95 |
\(\frac{1}{40}\)
|
\(\frac{1}{160}\)
| 9.11840e − 04 | 2.10 | 1.91874e − 03 | 2.09 | 4.64444e − 03 | 2.03 |
\(\frac{1}{80}\)
|
\(\frac{1}{640}\)
| 2.01991e − 04 | 2.17 | 3.97933e − 04 | 2.27 | 1.03486e − 03 | 2.17 |
h |
k
|
\(|\!|\!|u-u_{h}|\!|\!|\)
| Rate |
\(|\!|\!|R_{h}y-y_{h}|\!|\!|_{1}\)
| Rate |
\(|\!|\!|R_{h}p-p_{h}|\!|\!|_{1}\)
| Rate |
---|---|---|---|---|---|---|---|
\(\frac{1}{10}\)
|
\(\frac{1}{10}\)
| 2.45170e − 05 | – | 5.45644e − 04 | – | 3.56804e − 03 | – |
\(\frac{1}{20}\)
|
\(\frac{1}{40}\)
| 6.45526e − 06 | 1.93 | 1.29880e − 04 | 2.07 | 9.02173e − 04 | 1.98 |
\(\frac{1}{40}\)
|
\(\frac{1}{160}\)
| 1.72058e − 06 | 1.91 | 3.19495e − 05 | 2.02 | 2.26222e − 04 | 2.00 |
\(\frac{1}{80}\)
|
\(\frac{1}{640}\)
| 4.3535e − 07 | 1.98 | 7.85015e − 06 | 2.03 | 5.64788e − 05 | 2.00 |