State-dependent custom operators for block preconditioning

[tlroy]

Adds state-dependent custom Schur complement approximations for block preconditioners.

Users can provide a custom Schur operator directly, or define one from a weak form whose coefficients depend on the current nonlinear solution. Before each Newton linearization, the preconditioner rebuilds that operator using the latest block state, then uses it for the Schur solve. The PR also adds the weak-form helper API and tests for fixed and state-dependent custom Schur approximations.

[Copilot]

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Pull request overview

Adds support for state-dependent custom block-operator providers (including Schur complement approximations) so block preconditioners can rebuild their internal operators from the current nonlinear iterate before each Newton linearization. This extends the block preconditioner override API beyond fixed operators, wires state refresh into differentiable-numerics Newton solves, and adds weak-form helpers plus tests.

Changes:

• Introduces StateDependentSolver and state-refresh plumbing for block preconditioners and solver wrappers (SolverWithPreconditioner).
• Replaces fixed block-operator overrides with provider-based overrides (fixed and state-dependent) and adds factories to build them.
• Adds weak-form operator builder utilities for fixed/state-dependent overrides and expands tests (including a nonlinear mixed diffusion example).

File summaries

File	Description
src/smith/numerics/tests/test_equationsolver.cpp	Adds a test ensuring EquationSolver owns/attaches a caller-supplied preconditioner.
src/smith/numerics/tests/test_block_preconditioner_custom_operators.cpp	Updates tests to the new provider-override API and adds a state-dependent Schur provider test.
src/smith/numerics/state_dependent_solver.hpp	Adds a small interface for solvers that refresh internals from nonlinear state.
src/smith/numerics/solver_with_preconditioner.hpp	Makes the wrapper state-refreshable by forwarding updates to owned solver/preconditioner when applicable.
src/smith/numerics/equation_solver.hpp	Adds constructor overload for custom owned preconditioners; adds block sub-solver builder helper.
src/smith/numerics/equation_solver.cpp	Implements custom-preconditioner construction and factors linear-solver building; adds block sub-solver builder.
src/smith/numerics/CMakeLists.txt	Registers the new numerics header.
src/smith/numerics/block_preconditioner.hpp	Introduces provider-based override types/factories and updateForState API on block preconditioners.
src/smith/numerics/block_preconditioner.cpp	Implements fixed/state-dependent providers and refresh logic; updates Schur custom path to use providers.
src/smith/differentiable_numerics/weak_form_block_operator.hpp	Adds helper APIs to assemble weak-form Jacobians for use as block-preconditioner operators.
src/smith/differentiable_numerics/weak_form_block_operator.cpp	Implements weak-form operator assembly and state-dependent rebuild logic with DOF elimination support.
src/smith/differentiable_numerics/tests/test_weak_form_block_operator.cpp	Adds unit tests for fixed and state-dependent weak-form operator providers.
src/smith/differentiable_numerics/tests/test_state_dependent_preconditioner.cpp	Tests Newton-loop state refresh wiring for a state-dependent Schur operator provider.
src/smith/differentiable_numerics/tests/test_porous_heat_sink.cpp	Uses weak-form operator override for custom Schur approximation in an existing block solve test.
src/smith/differentiable_numerics/tests/test_nonlinear_mixed_diffusion.cpp	Adds an end-to-end nonlinear mixed diffusion test exercising state-dependent custom Schur preconditioning.
src/smith/differentiable_numerics/tests/test_mixed_poisson.cpp	Uses weak-form operator override for custom Schur approximation in an existing mixed Poisson test.
src/smith/differentiable_numerics/tests/CMakeLists.txt	Registers new differentiable-numerics tests.
src/smith/differentiable_numerics/nonlinear_block_solver.hpp	Adds ability to register a state-dependent solver to refresh before Jacobian assembly.
src/smith/differentiable_numerics/nonlinear_block_solver.cpp	Calls updateForState before Jacobian assembly; adds builder overload for custom preconditioners.
src/smith/differentiable_numerics/CMakeLists.txt	Adds the new weak-form operator source/header to the differentiable-numerics build.

Review details

• Files reviewed: 20/20 changed files
• Comments generated: 1
• Review effort level: Lite

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[tupek2]

Does this test add something important that the nonlinear mixed diffsusion doesn't test (or couldn't be easily modified to test). I worry about about us getting too much testing bloat. So consolidating or unifying, or extracting out common things from these tests would actually be preferred for me.

[tupek2]

Or, between test_mixed_poisson, mixed_diffusion, porous_heat_sink... what do we really need to be testing to be sufficiently confident without over testing early on an evolving capability.

[tlroy]

test_mixed_poisson is different from the other two since it is a saddle point problem and is a good starting point for other problems like that, where Schur complement approximations are important. nonlinear_diffusion is essentially a nonlinear version of the porous heat sink test. the linear test tests a bunch of the different factorizations, and the nonlinear version only looks at the custom operators. I guess those two could be combined to reduce the numbers of test files or lines of code, but it might increase test time since we'd do more nonlinear solves.