Adding essential BC elimination documentation #341
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| | Symbol | Description | | ||
| |--------|-------------| | ||
| | $D_0$ | The zero matrix | | ||
| | $D_1$ | Matrix with ones on the diagonal entries associates with essential DoFs and zeros elsewhere | |
| |--------|-------------| | ||
| | $D_0$ | The zero matrix | | ||
| | $D_1$ | Matrix with ones on the diagonal entries associates with essential DoFs and zeros elsewhere | | ||
| | $D_K$ | Matrix with $A_\{ii}$ on the diagonal entries associates with essential DoFs and zeros elsewhere | |
| | $D_K$ | Matrix with $A_\{ii}$ on the diagonal entries associates with essential DoFs and zeros elsewhere | | ||
| | $C_1$ | Matrix $(I - D_1)$ where $I$ is the identity matrix | | ||
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| Note that MFEM does actually not form these matrices. They are merely |
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does actually not --> does not actually
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| $$\begin{align\*} | ||
| A &\rightarrow C_1 A C_1 + D_p \\\\ | ||
| b &\rightarrow C_1 b - A D_1 x_\{bc} + D_p x_\{bc} |
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I think this is missing a bracket
| enumeration defined in | ||
| [linalg/operator.hpp](https://github.com/mfem/mfem/blob/master/linalg/operator.hpp#L47) | ||
| with values `DIAG_ZERO`, `DIAG_ONE`, or `DIAG_KEEP` which respectively | ||
| replace the diagonal entries with 0, 1, or leaves them unchanged. |
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Regarding the diagonal policy, this applies only to the legacy and serial assembly. Since HypreParMatrix doesn't support DIAG_KEEP, pbilinearform doesn't support it either. Maybe we should clarify it somewhere. MF/PA/FA operators also don't provide this option.
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| For a block system: | ||
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| $$\left(\begin{array}{3} |
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Since this equation (and the other 2 above) are not referred to in the text, their numbering can be omitted.
| b_1 \rightarrow & C_1 b_1 - A_\{11} D_1 x_\{bc} - A_\{12} D_1 y_\{bc} + D_p x_\{bc}, \\\\ | ||
| b_2 \rightarrow & C_1 b_2 - A_\{21}D_1 x_\{bc} - A_\{22} D_1 y_\{bc} + D_p y_\{bc}, \\\\ | ||
| b_3 \rightarrow & b_3 - A_\{31}D_1 x_\{bc} - A_\{32}D_1 y_\{bc} | ||
| \end{align\*}$$ |
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Same as above,
| a &\rightarrow C_1 a - A D_1 x_\{bc} + B D_1 y_\{bc} + D_p x_\{bc}\\\\ | ||
| b &\rightarrow C_1 b - A D_1 y_\{bc} - B D_1 x_\{bc} + D_p y_\{bc} | ||
| \end{align\*}$$ | ||
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I think this is also missing a bracket. It should be
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thanks for reviewing, @psocratis ! |
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This is an expanded and improved description of the essential BC elimination scheme mentioned in issue #5017.