Teko User's Guide

Teko is a Trilinos package for block and physics-based preconditioning. It provides tools to assemble and manipulate block linear operators, to decompose a fully coupled operator into its physical sub-blocks, and to build approximate inverses of those blocks using the full complement of Trilinos solvers and preconditioners (Ifpack2, MueLu, Amesos2, Belos, …). On top of that infrastructure Teko ships a library of generic block preconditioners (block Jacobi, block Gauss–Seidel, LU 2×2) and physics-based preconditioners for the Navier–Stokes equations (SIMPLE, LSC, PCD).

‍Teko means "fuse" in Greek — suggestive of what the package does: fusing multiple physics into a single preconditioner. The original motivation was preconditioners for magnetohydrodynamics and fusion-reactor simulation. For the theory behind the Navier–Stokes methods, see Cyr, Shadid & Tuminaro, Stabilization and Scalable Block Preconditioning for the Navier–Stokes Equations, JCP 2011.

This guide focuses on the parts of Teko that have historically been under-documented: the configuration options (the Teuchos::ParameterList / XML keys that control every preconditioner) and worked examples that show how the pieces fit together.

How to read this guide

Page	What it covers
Getting Started	Core concepts and a minimal end-to-end example: blocked operator → strategy → inverse → apply.
Configuration Model	The three-layer configuration model and the first complete Teko XML example. Start here for options.
The Inverse Library	How named inverses work and how they reference Ifpack2 / MueLu / Amesos2 / Belos.
Block Preconditioner Reference	Exhaustive parameter tables for every built-in block preconditioner.
Navier–Stokes Preconditioners	SIMPLE, LSC, PCD, and the operators they require from the application.
Examples	Walkthroughs of the in-tree example drivers, with run instructions.
Advanced Topics	The RequestHandler callback system, reordering, operator reuse, and writing your own factory.
Cookbook	Common blocking, solver-composition, reuse, debugging, and multiphysics recipes.

Glossary

• Block linear operator — a linear operator whose action is defined in terms of a grid of sub-blocks . In Teko these are Teko::BlockedLinearOp objects, ultimately Thyra::PhysicallyBlockedLinearOpBase.
• Strided blocking — the rule that maps the degrees of freedom of a monolithic operator onto blocks. For unknowns interleaved per node as [u v w p T], the strided-blocking string "3 1 1" groups the first three unknowns (velocity) into one block and puts p and T into their own blocks. See Configuration Model.
• Inverse factory (Teko::InverseFactory) — an object that, given a linear operator, produces a new operator approximating its inverse (a direct solve, a Krylov solve, a single-level preconditioner, or a Teko block preconditioner).
• Inverse library (Teko::InverseLibrary) — a registry of named inverse factories built from a parameter list. Names defined here are how one preconditioner references another. See The Inverse Library.
• Strategy — for the more sophisticated preconditioners (LSC, PCD, LU2x2) the Schur-complement approximation is factored out into a pluggable strategy object, selected by a "Strategy Name" parameter.
• RequestHandler (Teko::RequestHandler) — a callback mechanism through which a factory asks the application for operators it cannot assemble itself (e.g. a velocity mass matrix or a pressure Laplacian). See Advanced Topics.

The 30-second picture

Teko preconditioners are configured through nested Teuchos::ParameterLists (equivalently, XML). There are three layers, from the outside in:

1. Stratimikos layer. Teko registers itself as a Stratimikos preconditioner named "Teko". Its top-level keys say how to split the monolithic operator into blocks ("Strided Blocking", "Reorder Type") and which named inverse to use for the whole system ("Inverse Type").
2. Inverse Factory Library. A sublist named "Inverse Factory Library" defines every named inverse. Each entry has a mandatory "Type" that selects a backend — either a Stratimikos solver/preconditioner (Belos, Amesos2, Ifpack2, MueLu, Neumann Series, …) or a Teko block preconditioner ("Block Gauss-Seidel", "NS SIMPLE", …).
3. Per-factory options. Each Teko block preconditioner reads its own keys (which sub-solve to use for each block, relaxation factors, Schur-complement strategy, …). These reference other names defined in the library, so preconditioners nest arbitrarily.

The next page walks through this model with a complete example.