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\documentclass{article}
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\begin{document}
$ P_1 $
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$ \StiffElas $
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\[ \StiffElas_{m,l}=\int_{\DOMH} \Odv^t(\BasisFuncTwoD_m) \Ocv(\BasisFuncTwoD_l)dT, \ \forall (m,l)\in\ENS{1}{2\,\nq}^2,\]
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$ \BasisFuncTwoD_m $
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$ \Ocv=(\Occ_{xx},\Occ_{yy},\Occ_{xy})^t $
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$ \Odv=(\Odc_{xx},\Odc_{yy},2\Odc_{xy})^t $
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$ \Masse $
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\[ \Masse_{i,j}=\int_\DOMH \FoncBase_i(\q)\, \FoncBase_j(\q)\, d\q,\ \forall (i,j)\in{\ENS{1}{\nq}}^2 \]
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$ \FoncBase_i $
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$ \MasseF{w} $
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\[ \MasseF{w}_{i,j}=\int_\DOMH w(\q)\FoncBase_i(\q) \FoncBase_j(\q) d\q,\ \forall (i,j)\in\ENS{1}{\nq}^2 \]
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$ \Stiff $
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\[ \Stiff_{i,j}=\int_\DOMH \DOT{\GRAD\FoncBase_i(\q)}{\GRAD\FoncBase_j(\q)}d\q,\ \forall (i,j)\in{\ENS{1}{\nq}}^2\]
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$2\times\nq$
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${\q}(\il,j)$
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$\il$
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$j$
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$\il\in\{1,2\}$
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$j\in\ENS{1}{\nq}$
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$3\times\nme$
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$\me(\jl,k)$
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$\jl$
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$k$
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$\q$
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$\jl\in\{1,2,3\}$
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$k\in{\ENS{1}{\nme}}$
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$1\times\nq$
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$1\times\nme$
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$2\times\nbe$
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$\be(\il,l)$
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$l$
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$l\in{\ENS{1}{\nbe}}$
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$1\times\nbe$
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$\nq$
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$\nme$
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$\nbe$
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$lbe(j)$
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$\lbe(j)$
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$[0,1]\times[0,1]$
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$y=0$
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$x=1$
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$y=1$
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$x=0$
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$P_1$
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$3\times 3$
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\[AElem=\frac{area}{12}\left( \begin{array}{ccc} 2 &1 &1 \\ 1 &2 &1 \\ 1 &1 &2 \end{array} \right)\]
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$3\times 1$
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$2\times 1$
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$\Masse$
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\[\Masse_{i,j}=\int_\DOMH \FoncBase_i(\q)\, \FoncBase_j(\q)\, d\q,\ \forall (i,j)\in{\ENS{1}{\nq}}^2\]
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$\FoncBase_i$
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$\nq\times\nq$
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$\MasseF{w}$
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\[\MasseF{w}_{i,j}=\int_\DOMH w(\q)\FoncBase_i(\q) \FoncBase_j(\q) d\q,\ \forall (i,j)\in{\ENS{1}{\nq}}^2\]
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$w$
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$Tw(i)=w(\q^i),$
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$\forall i\in\ENS{1}{\nq}$
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$\Stiff$
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\[\Stiff_{i,j}=\int_\DOMH \DOT{\GRAD\FoncBase_i(\q)}{\GRAD\FoncBase_j(\q)}d\q,\ \forall (i,j)\in{\ENS{1}{\nq}}^2\]
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\[\StiffElas_{m,l}=\int_{\DOMH} \Odv^t(\BasisFuncTwoD_m) \Ocv(\BasisFuncTwoD_l)dT, \ \forall (m,l)\in\ENS{1}{2\,\nq}^2,\]
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$\BasisFuncTwoD_m$
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$\Ocv=(\Occ_{xx},\Occ_{yy},\Occ_{xy})^t$
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$\Odv=(\Odc_{xx},\Odc_{yy},2\Odc_{xy})^t$
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$2\nq\times 2\nq$
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\[\Masse_{i,j}=\int_\DOMH \FoncBase_i(\q) \FoncBase_j(\q) d\q,\ \forall (i,j)\in{\ENS{1}{\nq}}^2\]
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\[\MasseF{w}_{i,j}=\int_\DOMH w(\q)\;\FoncBase_i(\q)\; \FoncBase_j(\q)\; d\q,\ \forall (i,j)\in{\ENS{1}{\nq}}^2\]
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\[\Masse_{i,j}=\int_\DOMH \FoncBase_i(\q) \FoncBase_j(\q) d\q,\ \forall (i,j)\in\ENS{1}{\nq}^2\]
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\[\int_\DOM u(x,y) v(x,y) dxdy \approx \DOT{\Masse \vecb{U}}{\vecb{V}}\]
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$\vecb{U}_i=u(\q^i)$
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$\vecb{V}_i=v(\q^i)$
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$u$
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$v$
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\[|\int_\DOM u\,v -\Pi_h(u)\,\Pi_h(v)d\DOM| \leq C h^2\]
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\[\MasseF{w}_{i,j}=\int_\DOMH w(\q)\FoncBase_i(\q) \FoncBase_j(\q) d\q,\ \forall (i,j)\in\ENS{1}{\nq}^2\]
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\[\int_\DOM w(x,y) u(x,y) v(x,y) dxdy \approx \DOT{\MasseF{w} \vecb{U}}{\vecb{V}}\]
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\[|\int_\DOM u\,v\,w -\Pi_h(u)\,\Pi_h(v)\,\Pi_h(w)d\DOM| \leq C h^2\]
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\[\Stiff_{i,j}=\int_\DOMH \DOT{\GRAD\FoncBase_i(\q)}{\GRAD\FoncBase_j(\q)}d\q,\ \forall (i,j)\in\ENS{1}{\nq}^2\]
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\[\int_\DOM \DOT{\GRAD u(\q)}{\GRAD v(\q)}d\q \approx \DOT{\Stiff \vecb{U}}{\vecb{V}}\]
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\[|\int_\DOM \DOT{\GRAD u}{\GRAD v} -\DOT{\GRAD \Pi_h(u)}{\GRAD \Pi_h(v)}d\DOM| \leq C h^2\]
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\[\int_\DOM \Odv^t(\vecb{u}) \Ocv(\vecb{v})d\q \approx \DOT{\StiffElas \vecb{U}}{\vecb{V}}\]
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$\foncdefsmall{\vecb{u}=(u_1,u_2)}{\DOM}{\R^2}$
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$\foncdefsmall{\vecb{v}=(v_1,v_2)}{\DOM}{\R^2}$
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$\vecb{U}_{2i-1}=u_1(\q^i),$
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$\vecb{U}_{2i}=u_2(\q^i),$
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$\vecb{V}_{2i-1}=v_1(\q^i),$
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$\vecb{V}_{2i}=v_2(\q^i)$
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$\vecb{U}_{i}=u_1(\q^i),$
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$\vecb{U}_{i+\nq}=u_2(\q^i),$
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$\vecb{V}_{i}=v_1(\q^i),$
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$\vecb{V}_{i+\nq}=v_2(\q^i)$
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$\vecb{u}$
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\[|\int_\DOM \Odv^t(\vecb{u}) \Ocv(\vecb{v})d\q - \DOT{\StiffElas \vecb{U}}{\vecb{V}}| \leq C h^2\]
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$\LaTeX{}$
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\[\mbox{\input{\INPUTLATEXDIR/benchMatlabR2012b_MassP1.tex}} \]
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\[\mbox{\input{\INPUTLATEXDIR/benchMatlabR2012b_MassWP1.tex}} \]
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\[\mbox{\input{\INPUTLATEXDIR/benchMatlabR2012b_StiffP1.tex}} \]
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\[\mbox{\input{\INPUTLATEXDIR/benchMatlabR2012b_StiffElasP1.tex}} \]
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\[\mbox{\input{\INPUTLATEXDIR/benchOctave3.6.3_MassP1.tex}} \]
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\[\mbox{\input{\INPUTLATEXDIR/benchOctave3.6.3_MassWP1.tex}} \]
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\[\mbox{\input{\INPUTLATEXDIR/benchOctave3.6.3_StiffP1.tex}} \]
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\[\mbox{\input{\INPUTLATEXDIR/benchOctave3.6.3_StiffElasP1.tex}} \]
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\end{document}
