Linear algebra

All linear algebra functionality (coefficient vectors, matrices, linear solvers) works on top of a few C++ concepts defining a simple interface:

• terra::linalg::VectorLike for coefficient vectors
• terra::linalg::OperatorLike for linear operators (sparse matrices)
• terra::linalg::solvers::SolverLike for linear solvers

Note

I have not figured out how to display docstrings referring to concept requirements in Doxygen yet. Check the source code for more details.

Coefficient vectors

Coefficient vectors that implement the terra::linalg::VectorLike concept are (for hexahedral or wedge meshes)

• terra::linalg::VectorQ1Scalar (coefficient vector for scalar, linear finite element spaces)
• terra::linalg::VectorQ1Vec (coefficient vector for vector-valued, linear finite element spaces)
• terra::linalg::VectorQ1IsoQ2Q1 (coefficient vector the mixed, linear Q1-iso-Q2 / Q1 finite element space - stable pairing for Stokes)

Vector-vector operations are found as free functions in the terra::linalg namespace (see vector.hpp particularly). Use those functions instead of the members of the classes implementing the concept.

Note that the linear algebra functionality may hide implementation details of the coefficient vectors. For instance, the underlying grid may store duplicate nodes at subdomain boundaries. These duplicated nodes are hidden from the user through the terra::linalg::VectorLike interface and dealt with internally.

If you want to work directly on the underlying grid data, check out the kernels in grid_operations.hpp.

Operators

Matrices should implement the terra::linalg::OperatorLike concept. Note that the concept mostly only requires an implementation for a matrix-vector product. The concept is meant to primarily target matrix-free implementations of linear operators. Some concrete implementations are found in the namespace terra::fe::wedge::operators::shell.

Solvers

Linear solvers and preconditioners should implement the terra::linalg::solvers::SolverLike concept and rely on the arguments being coefficient vectors implementing the terra::linalg::VectorLike concept and operators implementing the terra::linalg::OperatorLike concept.

Preconditioners are just other classes that also implement the terra::linalg::solvers::SolverLike concept. They can then usually be passed to a solver as an optional argument.

See the namespace terra::linalg::solvers for concrete implementations of solvers and preconditioners.

Note that many solvers require passing in coefficient vectors that are required as temporary storage internally. To ensure that no large memory allocations are hidden from the user, the coefficient vectors have to be allocated manually in the application code and passed to the solver.

Example code

Have a look at the tests to see how to use the linear algebra functionality.