How to inverse block opeator matrix

[Raghavendra0117]

Dear all,
I am trying to solve two coupled equations,
C dT/dt + $\nabla . q$ = 0
q+ k $\nabla T$ = 0
over a domain $\Omega$. The discrete equations are
[Mt] [T]^{n+1} +[D] [ q ] ^{n+1}= [Mt] [T]^n
[D]^T [T]^{n+1} + [Mq] [q]^{n+1} = 0
where [T] is in L2 Space and [q] is in RT-space.
I am able to create a block operator/matrix that has [Mt], [D], [D]^T and [Mq] as its sub matrices.

It would be helpful if someone can suggest any existing method/routine that does the inversion
(with the help of some suitable pre-conditioner) of this block operator and operates on R.H.S to get [T]^{n+1} & [q]^{n+1}.

Regarding the preconditioner, i have gone through some of the examples in which the fespace has been supplied in the definition of preconditioner.
Eg:
if ( ads_a2 == NULL )
{
ParFiniteElementSpace *prec_fespace = (a2->StaticCondensationIsEnabled() ? a2->SCParFESpace() : &HDivFESpace);
ads_a2 = new HypreADS(*A2, prec_fespace);
}
if ( pcg_a2 == NULL )
{
pcg_a2 = new HyprePCG(*A2);
pcg_a2->SetTol(SOLVER_TOL);
pcg_a2->SetMaxIter(SOLVER_MAX_IT);
pcg_a2->SetPrintLevel(SOLVER_PRINT_LEVEL);
pcg_a2->SetPreconditioner(*ads_a2);
} [Ref: Joule miniapp]

I am confused for the case of mixed FE case, how should one proceed ?

Is it possible to use the hypre library's cg, gmres etc without help of the preconditioner?

Is there any example which does the inversion of block operator/matrix built on mixed FE spaces.

It would be helpful if someone can suggest any literature/document on solvers used in the MFEM library.

Thanks in advance,
Raghavendra Kollipara