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Add the definition of jump
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kinnala committed Jan 20, 2020
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Showing 1 changed file with 4 additions and 3 deletions.
7 changes: 4 additions & 3 deletions docs/examples/ex04.rst
Original file line number Diff line number Diff line change
Expand Up @@ -27,8 +27,8 @@ contact problem: find :math:`\boldsymbol{u}_i : \Omega_i \rightarrow
\boldsymbol{\sigma}_{i,t}(\boldsymbol{u}_i) &= \boldsymbol{0} && \text{on $\Gamma$,} \\
\sigma_{1,n}(\boldsymbol{u}_1(\boldsymbol{\gamma}(\boldsymbol{x})) - \sigma_{2,n}(\boldsymbol{u}_2)&=0 && \text{on $\Gamma$,} \\
[[u_n]] - g &\geq 0 && \text{on $\Gamma$,} \\
\sigma_{i,n}(\boldsymbol{u}_i)&\leq 0 && \text{on $\Gamma$,} \\
([[u_n]] - g) \sigma_{i,n}(\boldsymbol{u}_i) &= 0 && \text{on $\Gamma$,}
\sigma_{2,n}(\boldsymbol{u}_2)&\leq 0 && \text{on $\Gamma$,} \\
([[u_n]] - g) \sigma_{2,n}(\boldsymbol{u}_2) &= 0 && \text{on $\Gamma$,}
\end{aligned}
Expand All @@ -42,8 +42,9 @@ where
* :math:`g((x,y)) = 1 - \sqrt{1 - y^2}`,
* :math:`\boldsymbol{\gamma} : \Gamma \rightarrow \{ (x, y) \in \partial
\Omega_1 : x > 0 \}`, :math:`\boldsymbol{\gamma}((x,y)) = (g(x-1)+1, y)`,
* :math:`[[u_n]] = \boldsymbol{u}_1(\boldsymbol{\gamma}(\boldsymbol{x})) \cdot \boldsymbol{n} - \boldsymbol{u}_2(\boldsymbol{x}) \cdot \boldsymbol{n}`,

and the directions for evaluating :math:`\sigma_{1,n}` and
and the directions for evaluating :math:`[[u_n]]`, :math:`\sigma_{1,n}` and
:math:`\boldsymbol{\sigma}_{1,t}` are defined as :math:`\boldsymbol{n}=(1,0)`
and :math:`\boldsymbol{t}=(0,1)`.
This is a nonlinear problem since we do not know a priori which subset
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