Can you add comment lines between each line? Because just like this, it is hard to understand your code

When I replaced

 eqnFEM = Table[D[Xcomp[t, x][[i]], t] + Inactive[Div][Sum[((-D[[i,j]]*IdentityMatrix[1]).Inactive[Grad][Xcomp[[j]],{x}],{j,1,3}],{x}]==0, {i, 1, 3}];
bound1FEM = 
  Table[DirichletCondition[Xcomp[t, x][[i]] == list[[i]], 
    x == xbegin], {i, 1, 3}];
bound2FEM = 
  Table[DirichletCondition[Xcomp[t, x][[i]] == list[[i + 3]], 
    x == xend], {i, 1, 3}];

by

 eqnFEM = Table[D[Xcomp[t, x][[i]], t] + Sum[D[-D[[i,j]],x]*D[Xcomp[[j]],x],{j,1,3}], {i, 1, 3}]+Sum[-D[[i,j]]*D[Xcomp[[j]],{x,2}],{j,1,3}]=NeumannValue[0,x==xend], {i, 1, 3}];
     bound1FEM = 
       Table[DirichletCondition[Xcomp[t, x][[i]] == list[[i]], 
         x == xbegin], {i, 1, 3}];

I fixed the problem. I happy that it works, but I do not understand yet why it fixed the problem. Guide from Wolfram recommends first form, rather than second.

Also, in the results you obtain, were can clearly see that there is almost no diffusion over time. Is that what you expected?

About you first question: I just want to see difference betwen discretization by FEM and TPG. About you second question: If you re-write example as I wrote in previous message and write

z1Function = z1[[2]]
z1Function["ElementMesh"]

you will see that it is not none. There is Mesh, so FEM is used. If you write

{state} = 
 NDSolve`ProcessEquations[{eqnFEM, bound1FEM, bound2FEM, 
   initFEM}, {XMg, XFe, XCa}, {t, 0, 300000000}, {x, xbegin, xend}]

you will see

{NDSolve`StateData["<" 0. ">"]}

so t is considered as temporal variable and, hence, NDSolve automatically choose Method of Lines and FEM as method of discretization. I try to specify NDSolve use purely FEM, but calculation it too long. I think when NDSolve see Dirichlet Condition and initial problem at t=0. It automatically choose Method of Lines and FEM. If you re-write boundary condition without DirchletCondition, you will see that NDSolve automatically choose Method Of Lines and TPG as method of discretization.