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The symmetric two-grid error map from a pre-smoother and Galerkin correction
Definition
Let be symmetric positive definite, let be the Galerkin projector of The Galerkin coarse correction is an energy-orthogonal projector, and let be a linear pre-smoothing error map. For the energy inner product of The energy inner product and energy norm for a Hermitian positive-definite matrix, define the -adjoint of by
This is well-defined because is invertible. For all , , so it is the adjoint with respect to that energy inner product. The symmetric two-grid error map is
It maps an input error through pre-smoothing , exact coarse correction , and the -adjoint post-smoother.
Depends on
- Galerkin coarse matrices and the exact coarse-correction error map
- The Galerkin coarse correction is an energy-orthogonal projector
- The adjoint $T^*:W\to V$ is characterised by $\langle Tv,w\rangle_W=\langle v,T^*w\rangle_V$
- Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes
- The energy inner product $\langle u,v\rangle_A=\langle Au,v\rangle$ and energy norm $\|u\|_A$ for a Hermitian positive-definite matrix
Used by
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Sources
- Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Chapter 6 (standard reference, not scraped)
- Numerical Methods II, University of Heidelberg lecture notes, Chapter 10 (standard reference, not scraped)