Implement additive Schwarz Preconditioner

[bergbauer]

This PR implements a matrix-based additive Schwarz preconditioner where a sparse matrix is assembled first, then cell blocks are cut out and factorized using LU factorization. It can then be used as a preconditioner or as a smoother in multigrid.

[nfehn]

This is the only part of the code that is not fully clear in my opinion. In particular, I do not understand "renumber lexicographic".

It seems that the numbering needs to be consistent with read_dof_values()/begin_dof_values() used in the function apply_additive_schwarz_matrices(). Is there a way to make this more clear? Why isn't there a function like read_dof_indices() in analogy to read/begin_dof_values()? Personally, I am against using auto in such cases, but there might be different opinions.

[nfehn]

Maybe formulated differently: What is the reason that the code written here is / can not written as a matrix_free->celll_loop()?

[bergbauer]

FE_Q is numbered hierarchical but FEEvaluation expects lexicographic numbering. Should we add the same reference like here

exadg/include/exadg/operators/operator_base.cpp

Lines 1894 to 1902 in ee073c7

	if(not is_dg)
	{
	// in the case of CG: shape functions are not ordered lexicographically
	// see (https://www.dealii.org/8.5.1/doxygen/deal.II/classFE__Q.html)
	// so we have to fix the order
	auto temp = dof_indices;
	for(unsigned int j = 0; j < dof_indices.size(); j++)
	dof_indices[j] = temp[matrix_free.get_shape_info().lexicographic_numbering[j]];
	}

?

FEEvaluation::get_dof_indices() would indeed sometimes make sense, but we usually don't access the dof indices in performance-critical places.

We could also write this as a cell loop but I tend to write out the for loop over cell batches when we don't work on dof vectors.

[nfehn]

I am somewhat confused. according to your explanations and https://dealii.org/current/doxygen/deal.II/structinternal_1_1MatrixFreeFunctions_1_1ShapeInfo.html#abad0471d12c4f07f3d83271377e24231, I would expect the code to read

lapack_matrix.set(lex[i], lex[j], overlapped_cell_matrix[i][j]);

because lapack_matrix uses lexicographic numerbing and overlapped_cell_matrix hierarchical numbering, and because shape_info::lexicographic_numbering is a "Renumbering from deal.II's numbering of cell degrees of freedom to lexicographic numbering used inside FEEvaluation"?

Generally, I think one should be able to get dof indices in the meaning of / consistent with integrator.read/begin_dof_values().

[nfehn]

I discussed this briefly with @peterrum. We think both the name shape_info.lexicograhpic_numbering and the deal.II docu is confusing. It should be lexicographic_to_hierarchical_numbering in my opinion. The current implementation of this PR would then be correct apart from renaming.

[bergbauer]

The documentation is really confusing, I agree. The renumbering in correct as it currently is.

[bergbauer]

I renamed the variable but the docu has to be fixed in deal.II. 13e5d64

[nfehn]

The code looks overall good from my perspective. I raised some questions, mainly to improve my own understanding of the code (or MatrixFree).

[nfehn]

Personally, I am a fan to keep code with lambdas as simple as possible, which is why I would probably define the lambda outside (e.g. cellwise_operation). But I am fine with the present version as well.

[nfehn]

looks good!