Coupled Finite Elements

$ \newcommand{\Ua}{\mathrm{a}} \newcommand{\Ub}{\mathrm{b}} \newcommand{\Uc}{\mathrm{c}} \newcommand{\Ud}{\mathrm{d}} \newcommand{\Ue}{\mathrm{e}} \newcommand{\Uf}{\mathrm{f}} \newcommand{\Ug}{\mathrm{g}} \newcommand{\Uh}{\mathrm{h}} \newcommand{\Ui}{\mathrm{i}} \newcommand{\Uj}{\mathrm{j}} \newcommand{\Uk}{\mathrm{k}} \newcommand{\Ul}{\mathrm{l}} \newcommand{\Um}{\mathrm{m}} \newcommand{\Un}{\mathrm{n}} \newcommand{\Uo}{\mathrm{o}} \newcommand{\Up}{\mathrm{p}} \newcommand{\Uq}{\mathrm{q}} \newcommand{\Ur}{\mathrm{r}} \newcommand{\Us}{\mathrm{s}} \newcommand{\Ut}{\mathrm{t}} \newcommand{\Uu}{\mathrm{u}} \newcommand{\Uv}{\mathrm{v}} \newcommand{\Uw}{\mathrm{w}} \newcommand{\Ux}{\mathrm{x}} \newcommand{\Uy}{\mathrm{y}} \newcommand{\Uz}{\mathrm{z}} \newcommand{\UA}{\mathrm{A}} \newcommand{\UB}{\mathrm{B}} \newcommand{\UC}{\mathrm{C}} \newcommand{\UD}{\mathrm{D}} \newcommand{\UE}{\mathrm{E}} \newcommand{\UF}{\mathrm{F}} \newcommand{\UG}{\mathrm{G}} \newcommand{\UH}{\mathrm{H}} \newcommand{\UI}{\mathrm{I}} \newcommand{\UJ}{\mathrm{J}} \newcommand{\UK}{\mathrm{K}} \newcommand{\UL}{\mathrm{L}} \newcommand{\UM}{\mathrm{M}} \newcommand{\UN}{\mathrm{N}} \newcommand{\UO}{\mathrm{O}} \newcommand{\UP}{\mathrm{P}} \newcommand{\UQ}{\mathrm{Q}} \newcommand{\UR}{\mathrm{R}} \newcommand{\US}{\mathrm{S}} \newcommand{\UT}{\mathrm{T}} \newcommand{\UU}{\mathrm{U}} \newcommand{\UV}{\mathrm{V}} \newcommand{\UW}{\mathrm{W}} \newcommand{\UX}{\mathrm{X}} \newcommand{\UY}{\mathrm{Y}} \newcommand{\UZ}{\mathrm{Z}} % \newcommand{\Uzero }{\mathrm{0}} \newcommand{\Uone }{\mathrm{1}} \newcommand{\Utwo }{\mathrm{2}} \newcommand{\Uthree}{\mathrm{3}} \newcommand{\Ufour }{\mathrm{4}} \newcommand{\Ufive }{\mathrm{5}} \newcommand{\Usix }{\mathrm{6}} \newcommand{\Useven}{\mathrm{7}} \newcommand{\Ueight}{\mathrm{8}} \newcommand{\Unine }{\mathrm{9}} % \newcommand{\Ja}{\mathit{a}} \newcommand{\Jb}{\mathit{b}} \newcommand{\Jc}{\mathit{c}} \newcommand{\Jd}{\mathit{d}} \newcommand{\Je}{\mathit{e}} \newcommand{\Jf}{\mathit{f}} \newcommand{\Jg}{\mathit{g}} \newcommand{\Jh}{\mathit{h}} \newcommand{\Ji}{\mathit{i}} \newcommand{\Jj}{\mathit{j}} \newcommand{\Jk}{\mathit{k}} \newcommand{\Jl}{\mathit{l}} \newcommand{\Jm}{\mathit{m}} \newcommand{\Jn}{\mathit{n}} \newcommand{\Jo}{\mathit{o}} \newcommand{\Jp}{\mathit{p}} \newcommand{\Jq}{\mathit{q}} \newcommand{\Jr}{\mathit{r}} \newcommand{\Js}{\mathit{s}} \newcommand{\Jt}{\mathit{t}} \newcommand{\Ju}{\mathit{u}} \newcommand{\Jv}{\mathit{v}} \newcommand{\Jw}{\mathit{w}} \newcommand{\Jx}{\mathit{x}} \newcommand{\Jy}{\mathit{y}} \newcommand{\Jz}{\mathit{z}} \newcommand{\JA}{\mathit{A}} \newcommand{\JB}{\mathit{B}} \newcommand{\JC}{\mathit{C}} \newcommand{\JD}{\mathit{D}} \newcommand{\JE}{\mathit{E}} \newcommand{\JF}{\mathit{F}} \newcommand{\JG}{\mathit{G}} \newcommand{\JH}{\mathit{H}} \newcommand{\JI}{\mathit{I}} \newcommand{\JJ}{\mathit{J}} \newcommand{\JK}{\mathit{K}} \newcommand{\JL}{\mathit{L}} \newcommand{\JM}{\mathit{M}} \newcommand{\JN}{\mathit{N}} \newcommand{\JO}{\mathit{O}} \newcommand{\JP}{\mathit{P}} \newcommand{\JQ}{\mathit{Q}} \newcommand{\JR}{\mathit{R}} \newcommand{\JS}{\mathit{S}} \newcommand{\JT}{\mathit{T}} \newcommand{\JU}{\mathit{U}} \newcommand{\JV}{\mathit{V}} \newcommand{\JW}{\mathit{W}} \newcommand{\JX}{\mathit{X}} \newcommand{\JY}{\mathit{Y}} \newcommand{\JZ}{\mathit{Z}} % \newcommand{\Jzero }{\mathit{0}} \newcommand{\Jone }{\mathit{1}} \newcommand{\Jtwo }{\mathit{2}} \newcommand{\Jthree}{\mathit{3}} \newcommand{\Jfour }{\mathit{4}} \newcommand{\Jfive }{\mathit{5}} \newcommand{\Jsix }{\mathit{6}} \newcommand{\Jseven}{\mathit{7}} \newcommand{\Jeight}{\mathit{8}} \newcommand{\Jnine }{\mathit{9}} % \newcommand{\BA}{\boldsymbol{A}} \newcommand{\BB}{\boldsymbol{B}} \newcommand{\BC}{\boldsymbol{C}} \newcommand{\BD}{\boldsymbol{D}} \newcommand{\BE}{\boldsymbol{E}} \newcommand{\BF}{\boldsymbol{F}} \newcommand{\BG}{\boldsymbol{G}} \newcommand{\BH}{\boldsymbol{H}} \newcommand{\BI}{\boldsymbol{I}} \newcommand{\BJ}{\boldsymbol{J}} \newcommand{\BK}{\boldsymbol{K}} \newcommand{\BL}{\boldsymbol{L}} \newcommand{\BM}{\boldsymbol{M}} \newcommand{\BN}{\boldsymbol{N}} \newcommand{\BO}{\boldsymbol{O}} \newcommand{\BP}{\boldsymbol{P}} \newcommand{\BQ}{\boldsymbol{Q}} \newcommand{\BR}{\boldsymbol{R}} \newcommand{\BS}{\boldsymbol{S}} \newcommand{\BT}{\boldsymbol{T}} \newcommand{\BU}{\boldsymbol{U}} \newcommand{\BV}{\boldsymbol{V}} \newcommand{\BW}{\boldsymbol{W}} \newcommand{\BX}{\boldsymbol{X}} \newcommand{\BY}{\boldsymbol{Y}} \newcommand{\BZ}{\boldsymbol{Z}} \newcommand{\Ba}{\boldsymbol{a}} \newcommand{\Bb}{\boldsymbol{b}} \newcommand{\Bc}{\boldsymbol{c}} \newcommand{\Bd}{\boldsymbol{d}} \newcommand{\Be}{\boldsymbol{e}} \newcommand{\Bf}{\boldsymbol{f}} \newcommand{\Bg}{\boldsymbol{g}} \newcommand{\Bh}{\boldsymbol{h}} \newcommand{\Bi}{\boldsymbol{i}} \newcommand{\Bj}{\boldsymbol{j}} \newcommand{\Bk}{\boldsymbol{k}} \newcommand{\Bl}{\boldsymbol{l}} \newcommand{\Bm}{\boldsymbol{m}} \newcommand{\Bn}{\boldsymbol{n}} \newcommand{\Bo}{\boldsymbol{o}} \newcommand{\Bp}{\boldsymbol{p}} \newcommand{\Bq}{\boldsymbol{q}} \newcommand{\Br}{\boldsymbol{r}} \newcommand{\Bs}{\boldsymbol{s}} \newcommand{\Bt}{\boldsymbol{t}} \newcommand{\Bu}{\boldsymbol{u}} \newcommand{\Bv}{\boldsymbol{v}} \newcommand{\Bw}{\boldsymbol{w}} \newcommand{\Bx}{\boldsymbol{x}} \newcommand{\By}{\boldsymbol{y}} \newcommand{\Bz}{\boldsymbol{z}} % \newcommand{\Bzero }{\boldsymbol{0}} \newcommand{\Bone }{\boldsymbol{1}} \newcommand{\Btwo }{\boldsymbol{2}} \newcommand{\Bthree}{\boldsymbol{3}} \newcommand{\Bfour }{\boldsymbol{4}} \newcommand{\Bfive }{\boldsymbol{5}} \newcommand{\Bsix }{\boldsymbol{6}} \newcommand{\Bseven}{\boldsymbol{7}} \newcommand{\Beight}{\boldsymbol{8}} \newcommand{\Bnine }{\boldsymbol{9}} % \newcommand{\Balpha }{\boldsymbol{\alpha} } \newcommand{\Bbeta }{\boldsymbol{\beta} } \newcommand{\Bgamma }{\boldsymbol{\gamma} } \newcommand{\Bdelta }{\boldsymbol{\delta} } \newcommand{\Bepsilon}{\boldsymbol{\epsilon} } \newcommand{\Bvareps }{\boldsymbol{\varepsilon} } \newcommand{\Bvarepsilon}{\boldsymbol{\varepsilon}} \newcommand{\Bzeta }{\boldsymbol{\zeta} } \newcommand{\Beta }{\boldsymbol{\eta} } \newcommand{\Btheta }{\boldsymbol{\theta} } \newcommand{\Bvarthe }{\boldsymbol{\vartheta} } \newcommand{\Biota }{\boldsymbol{\iota} } \newcommand{\Bkappa }{\boldsymbol{\kappa} } \newcommand{\Blambda }{\boldsymbol{\lambda} } \newcommand{\Bmu }{\boldsymbol{\mu} } \newcommand{\Bnu }{\boldsymbol{\nu} } \newcommand{\Bxi }{\boldsymbol{\xi} } \newcommand{\Bpi }{\boldsymbol{\pi} } \newcommand{\Brho }{\boldsymbol{\rho} } \newcommand{\Bvrho }{\boldsymbol{\varrho} } \newcommand{\Bsigma }{\boldsymbol{\sigma} } \newcommand{\Bvsigma }{\boldsymbol{\varsigma} } \newcommand{\Btau }{\boldsymbol{\tau} } \newcommand{\Bupsilon}{\boldsymbol{\upsilon} } \newcommand{\Bphi }{\boldsymbol{\phi} } \newcommand{\Bvarphi }{\boldsymbol{\varphi} } \newcommand{\Bchi }{\boldsymbol{\chi} } \newcommand{\Bpsi }{\boldsymbol{\psi} } \newcommand{\Bomega }{\boldsymbol{\omega} } \newcommand{\BGamma }{\boldsymbol{\Gamma} } \newcommand{\BDelta }{\boldsymbol{\Delta} } \newcommand{\BTheta }{\boldsymbol{\Theta} } \newcommand{\BLambda }{\boldsymbol{\Lambda} } \newcommand{\BXi }{\boldsymbol{\Xi} } \newcommand{\BPi }{\boldsymbol{\Pi} } \newcommand{\BSigma }{\boldsymbol{\Sigma} } \newcommand{\BUpsilon}{\boldsymbol{\Upsilon} } \newcommand{\BPhi }{\boldsymbol{\Phi} } \newcommand{\BPsi }{\boldsymbol{\Psi} } \newcommand{\BOmega }{\boldsymbol{\Omega} } % \newcommand{\IA}{\mathbb{A}} \newcommand{\IB}{\mathbb{B}} \newcommand{\IC}{\mathbb{C}} \newcommand{\ID}{\mathbb{D}} \newcommand{\IE}{\mathbb{E}} \newcommand{\IF}{\mathbb{F}} \newcommand{\IG}{\mathbb{G}} \newcommand{\IH}{\mathbb{H}} \newcommand{\II}{\mathbb{I}} \renewcommand{\IJ}{\mathbb{J}} \newcommand{\IK}{\mathbb{K}} \newcommand{\IL}{\mathbb{L}} \newcommand{\IM}{\mathbb{M}} \newcommand{\IN}{\mathbb{N}} \newcommand{\IO}{\mathbb{O}} \newcommand{\IP}{\mathbb{P}} \newcommand{\IQ}{\mathbb{Q}} \newcommand{\IR}{\mathbb{R}} \newcommand{\IS}{\mathbb{S}} \newcommand{\IT}{\mathbb{T}} \newcommand{\IU}{\mathbb{U}} \newcommand{\IV}{\mathbb{V}} \newcommand{\IW}{\mathbb{W}} \newcommand{\IX}{\mathbb{X}} \newcommand{\IY}{\mathbb{Y}} \newcommand{\IZ}{\mathbb{Z}} % \newcommand{\FA}{\mathsf{A}} \newcommand{\FB}{\mathsf{B}} \newcommand{\FC}{\mathsf{C}} \newcommand{\FD}{\mathsf{D}} \newcommand{\FE}{\mathsf{E}} \newcommand{\FF}{\mathsf{F}} \newcommand{\FG}{\mathsf{G}} \newcommand{\FH}{\mathsf{H}} \newcommand{\FI}{\mathsf{I}} \newcommand{\FJ}{\mathsf{J}} \newcommand{\FK}{\mathsf{K}} \newcommand{\FL}{\mathsf{L}} \newcommand{\FM}{\mathsf{M}} \newcommand{\FN}{\mathsf{N}} \newcommand{\FO}{\mathsf{O}} \newcommand{\FP}{\mathsf{P}} \newcommand{\FQ}{\mathsf{Q}} \newcommand{\FR}{\mathsf{R}} \newcommand{\FS}{\mathsf{S}} \newcommand{\FT}{\mathsf{T}} \newcommand{\FU}{\mathsf{U}} \newcommand{\FV}{\mathsf{V}} \newcommand{\FW}{\mathsf{W}} \newcommand{\FX}{\mathsf{X}} \newcommand{\FY}{\mathsf{Y}} \newcommand{\FZ}{\mathsf{Z}} \newcommand{\Fa}{\mathsf{a}} \newcommand{\Fb}{\mathsf{b}} \newcommand{\Fc}{\mathsf{c}} \newcommand{\Fd}{\mathsf{d}} \newcommand{\Fe}{\mathsf{e}} \newcommand{\Ff}{\mathsf{f}} \newcommand{\Fg}{\mathsf{g}} \newcommand{\Fh}{\mathsf{h}} \newcommand{\Fi}{\mathsf{i}} \newcommand{\Fj}{\mathsf{j}} \newcommand{\Fk}{\mathsf{k}} \newcommand{\Fl}{\mathsf{l}} \newcommand{\Fm}{\mathsf{m}} \newcommand{\Fn}{\mathsf{n}} \newcommand{\Fo}{\mathsf{o}} \newcommand{\Fp}{\mathsf{p}} \newcommand{\Fq}{\mathsf{q}} \newcommand{\Fr}{\mathsf{r}} \newcommand{\Fs}{\mathsf{s}} \newcommand{\Ft}{\mathsf{t}} \newcommand{\Fu}{\mathsf{u}} \newcommand{\Fv}{\mathsf{v}} \newcommand{\Fw}{\mathsf{w}} \newcommand{\Fx}{\mathsf{x}} \newcommand{\Fy}{\mathsf{y}} \newcommand{\Fz}{\mathsf{z}} % \newcommand{\Fzero }{\mathsf{0}} \newcommand{\Fone }{\mathsf{1}} \newcommand{\Ftwo }{\mathsf{2}} \newcommand{\Fthree}{\mathsf{3}} \newcommand{\Ffour }{\mathsf{4}} \newcommand{\Ffive }{\mathsf{5}} \newcommand{\Fsix }{\mathsf{6}} \newcommand{\Fseven}{\mathsf{7}} \newcommand{\Feight}{\mathsf{8}} \newcommand{\Fnine }{\mathsf{9}} % \newcommand{\CA}{\mathcal{A}} \newcommand{\CB}{\mathcal{B}} \newcommand{\CC}{\mathcal{C}} \newcommand{\CD}{\mathcal{D}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}} \newcommand{\CG}{\mathcal{G}} \newcommand{\CH}{\mathcal{H}} \newcommand{\CI}{\mathcal{I}} \newcommand{\CJ}{\mathcal{J}} \newcommand{\CK}{\mathcal{K}} \newcommand{\CL}{\mathcal{L}} \newcommand{\CM}{\mathcal{M}} \newcommand{\CN}{\mathcal{N}} \newcommand{\CO}{\mathcal{O}} \newcommand{\CP}{\mathcal{P}} \newcommand{\CQ}{\mathcal{Q}} \newcommand{\CR}{\mathcal{R}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CT}{\mathcal{T}} \newcommand{\CU}{\mathcal{U}} \newcommand{\CV}{\mathcal{V}} \newcommand{\CW}{\mathcal{W}} \newcommand{\CX}{\mathcal{X}} \newcommand{\CY}{\mathcal{Y}} \newcommand{\CZ}{\mathcal{Z}} % \newcommand{\KA}{\mathfrak{A}} \newcommand{\KB}{\mathfrak{B}} \newcommand{\KC}{\mathfrak{C}} \newcommand{\KD}{\mathfrak{D}} \newcommand{\KE}{\mathfrak{E}} \newcommand{\KF}{\mathfrak{F}} \newcommand{\KG}{\mathfrak{G}} \newcommand{\KH}{\mathfrak{H}} \newcommand{\KI}{\mathfrak{I}} \newcommand{\KJ}{\mathfrak{J}} \newcommand{\KK}{\mathfrak{K}} \newcommand{\KL}{\mathfrak{L}} \newcommand{\KM}{\mathfrak{M}} \newcommand{\KN}{\mathfrak{N}} \newcommand{\KO}{\mathfrak{O}} \newcommand{\KP}{\mathfrak{P}} \newcommand{\KQ}{\mathfrak{Q}} \newcommand{\KR}{\mathfrak{R}} \newcommand{\KS}{\mathfrak{S}} \newcommand{\KT}{\mathfrak{T}} \newcommand{\KU}{\mathfrak{U}} \newcommand{\KV}{\mathfrak{V}} \newcommand{\KW}{\mathfrak{W}} \newcommand{\KX}{\mathfrak{X}} \newcommand{\KY}{\mathfrak{Y}} \newcommand{\KZ}{\mathfrak{Z}} \newcommand{\Ka}{\mathfrak{a}} \newcommand{\Kb}{\mathfrak{b}} \newcommand{\Kc}{\mathfrak{c}} \newcommand{\Kd}{\mathfrak{d}} \newcommand{\Ke}{\mathfrak{e}} \newcommand{\Kf}{\mathfrak{f}} \newcommand{\Kg}{\mathfrak{g}} \newcommand{\Kh}{\mathfrak{h}} \newcommand{\Ki}{\mathfrak{i}} \newcommand{\Kj}{\mathfrak{j}} \newcommand{\Kk}{\mathfrak{k}} \newcommand{\Kl}{\mathfrak{l}} \newcommand{\Km}{\mathfrak{m}} \newcommand{\Kn}{\mathfrak{n}} \newcommand{\Ko}{\mathfrak{o}} \newcommand{\Kp}{\mathfrak{p}} \newcommand{\Kq}{\mathfrak{q}} \newcommand{\Kr}{\mathfrak{r}} \newcommand{\Ks}{\mathfrak{s}} \newcommand{\Kt}{\mathfrak{t}} \newcommand{\Ku}{\mathfrak{u}} \newcommand{\Kv}{\mathfrak{v}} \newcommand{\Kw}{\mathfrak{w}} \newcommand{\Kx}{\mathfrak{x}} \newcommand{\Ky}{\mathfrak{y}} \newcommand{\Kz}{\mathfrak{z}} % \newcommand{\Kzero }{\mathfrak{0}} \newcommand{\Kone }{\mathfrak{1}} \newcommand{\Ktwo }{\mathfrak{2}} \newcommand{\Kthree}{\mathfrak{3}} \newcommand{\Kfour }{\mathfrak{4}} \newcommand{\Kfive }{\mathfrak{5}} \newcommand{\Ksix }{\mathfrak{6}} \newcommand{\Kseven}{\mathfrak{7}} \newcommand{\Keight}{\mathfrak{8}} \newcommand{\Knine }{\mathfrak{9}} % $

In this post, I’ll introduce the FE formulation of a generalized linear and coupled weak form. Said weak formulation has the form

Find $u\in V_1$, $y\in V_2$ such that
\[\begin{equation} \begin{alignedat}{3} a(u, v) &+ b(y, v) &&= c(v) \\ d(u, w) &+ e(y, w) &&= f(w) \\ \end{alignedat} \label{eq:coupledweakform1} \end{equation}\]
for all $v\in V_1$, $w \in V_2$ where $a(\cdot, \cdot): V_1\times V_1 \to \IR$, $b(\cdot, \cdot): V_2\times V_1 \to \IR$, $d(\cdot, \cdot): V_1\times V_2 \to \IR$, $e(\cdot, \cdot): V_2\times V_2 \to \IR$ are bilinear forms and $c(\cdot): V_1\to \IR$, $f(\cdot): V_2\to \IR$ are linear forms.

Here, the objective is to solve for the two unknown functions $u$ and $y$. One can also imagine an arbitrary degree of coupling between $n$ variables with $n$ equations.

We introduce the following discretizations

\[\begin{equation} \begin{alignedat}{3} u_h &= \suml{J=1}{n_n^1} u^J N^J \qquad\qquad & v_h &= \suml{I=1}{n_n^1} u^I N^I \qquad\qquad & u_h, v_h\in V_{h1} \\ y_h &= \suml{L=1}{n_n^2} u^L N^L & w_h &= \suml{K=1}{n_n^2} u^K N^K & y_h, w_h\in V_{h2} \\ \end{alignedat} \end{equation}\]

where the corresponding number of shape functions are $n_n^1$ and $n_n^2$, respectively.

Substituting the discretizations in \eqref{eq:coupledweakform1}, we obtain two linear systems of equations

\[\begin{equation} \begin{alignedat}{3} \suml{J=1}{n_n^1} a(N^J, N^I) \,u^J &+ \suml{L=1}{n_n^2} b(N^L, N^I) \, y^L &&= c(N^I) \\ \suml{J=1}{n_n^1} d(N^J, N^K) \,u^J &+ \suml{L=1}{n_n^2} e(N^L, N^K) \, y^L &&= f(N^K) \\ \end{alignedat} \end{equation}\]

for $I=1,\dots,n_n^1$ and $K=1,\dots,n_n^2$.

We write this system as

\[\begin{equation} \boxed{ \begin{alignedat}{3} \BA \Bu &+ \BB\By &&= \Bc \\ \BD \Bu &+ \BE\By &&= \Bf \\ \end{alignedat} \eqor \begin{bmatrix} \BA & \BB \\ \BD & \BE \end{bmatrix} \begin{bmatrix} \Bu \\ \By \end{bmatrix} = \begin{bmatrix} \Bc \\ \Bf \end{bmatrix} } \label{eq:coupledsystem1} \end{equation}\]

where the components of given matrices and vectors are defined as

\[\begin{equation} \begin{alignedat}{6} A^{I\!J} &:= a(N^J, N^I) \qquad & B^{I\!L} &:= b(N^L, N^I) \qquad & c^{I} &:= c(N^I)\\ D^{K\!J} &:= d(N^J, N^K) & E^{K\!L} &:= e(N^L, N^K) & f^{K} &:= f(N^K)\\ \end{alignedat} \end{equation}\]

Solution of \eqref{eq:coupledsystem1} yields the unknown vectors $\Bu$ and $\By$.