Linear Solvers¶
Once you have assembled the relevant operators, and have created a grid function containing the relevant right-hand-side data, you will need to solve your linear system.
Full documentation of Bempp linear solvers can be found on Read the Docs.
Direct Solvers¶
Direct solvers compute the solution of a linear system by (usually indirectly) computing the inverse of the matrix.
SciPy’s direct LU solver is wrapped in the function bempp.api.linalg.lu. This can be used with:
solution = bempp.api.linalg.lu(operator, grid_fun)
Direct solvers should only be used if the operator has been assembled in dense mode.
Iterative Solvers¶
Iterative solvers solve a linear system iteratively: steps are repeated to achieve better approximations of the solution. For well-condtioned matrices, iterative solvers can achieve fast convergence, so very good approximations of the solution can be achieved in just a few iterations.
SciPy’s CG and GMRes iterative solvers are wrapped in the bempp.api.linalg submodule. These
can be used with:
solution, info = bempp.api.linalg.cg(operator, grid_fun)
solution, info = bempp.api.linalg.gmres(opreator, grid_fun)
These solvers take a number of optional arguments:
Argument |
Description |
Default |
|---|---|---|
|
The tolerance the solver should aim for |
|
|
The maximum number of iterations |
No maximum |
|
If |
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|
If |
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|
If |
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By default, Bempp will use the weak form discretisation of the operator and the coefficients
of the grid function when using an iterative solver.
If use_strong_form is set to True, Bempp will use the strong form discretisation of the operator
and the projections of the grid function onto the range of the operator. This is equivalent
to applying a mass matrix preconditioner to the problem and often leads to a lower iteration count.