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How can I reproduce the following drawing using LaTeX?

I suspect that TikZ has the right instruments to do it, but I'm not familiar with the package. Indeed, the most I can do with it is drawing basic shapes, like

usepackage{tikz}

begin{tikzpicture}

    draw[step=1cm,gray,very thin] (-1.9,-1.9) grid (5.9,5.9);

end{tikzpicture}

or

usepackage{tikz}

begin{tikzpicture}

      draw (0,0) -- (4,0) -- (4,4) -- (0,4) -- (0,0);

end{tikzpicture}

but I don't know how to combine them to create the desired end product and how to add labels.

A solution with pstricks:

documentclass[svgnames]{standalone}

usepackage{pst-node}

usepackage{auto-pst-pdf} 



begin{document}



psset{linejoin=1, dash=3.8pt 3pt, dimen=middle}

begin{pspicture}(-0.6, -0.9)(6.2, 4.2)

psframe(4,4)

psline[linestyle=dashed](0, 1.2)(2, 1.2)(2,2.8)(0,2.8)

psframe[fillstyle =solid,fillcolor=Silver](1.4,0)(2.6,1.2)

dotnodes(0,2){F}(2,2){G}(2,0){H}



uput[l](F){$F_j$}

rput[l](5,0.5){$x_n=0$}

rput(0.7, 1.9){$Q_{F_j}$}

pnodes(3.2, 2.7){E}(2.8,-0.4){Q}

uput{3pt}[110](E){$e_n$,}

rput(2,0.6){Rnode{R}{$R_{1/2}$}}

rput[tl](Q){,$Q_{1/2}(1/4e_n)$}

psset{linewidth=0.4pt, arrows=->}

ncline[nodesepB=2pt]{Q}{R}

ncdiagg[angleA=180]{E}{G}

end{pspicture}



end{document} 

There are infinite ways to do it, here's one with TikZ:

documentclass[border=1cm]{standalone}

usepackage{tikz}

usetikzlibrary{positioning, arrows.meta}

tikzset{quadro/.style={draw,              

            rectangle,

            inner sep=0pt,

            minimum height=#1,

            text width=#1},

        point/.style={draw,             

            circle,

            inner sep=1.5pt,

            fill=black}}

begin{document}

    begin{tikzpicture}[font=Large, thick]

        node[quadro={8cm}]

            (mysquare)

            {};

        node[quadro={4cm}, draw=none,  anchor=west, text centered] (mydashed) at (mysquare.west) {$Q_{F_{j}}$};

        draw[dashed] (mydashed.south west) -- (mydashed.south east) -- (mydashed.north east) -- (mydashed.north west);

        node[quadro={2cm}, fill=lightgray, anchor=south, text centered] at (mysquare.south) (graysquare) {$R_{1/2}$};

        node[point] {};

        node[point, label=left:{$F_{j}$}] at (mysquare.west) {}; 

        node[point] at (mysquare.south) {};

        node[below left=.3cm and -.7cm of mysquare.south east] (descr1) {$Q_{1/2}(1/4e_{n})$};

        draw[-Stealth] (descr1.west) -- ([shift={(.25cm,.5cm)}]graysquare.south); 

        node[above right= .1cm and .1cm of mysquare.south east, anchor=south west]  {$x_{n}=0$};

        node[below left= 2cm and 1.5cm of mysquare.north east] (descr2) {$e_{n}$};

        draw[-Stealth, shorten >=0.2cm] (descr2.south east) -- (descr2.south west) -- (mydashed.east);

    end{tikzpicture}

end{document}