Alphabeta MathHow statement and proof provenance work
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The Hilbert-space discrete energy identity
Statement
For every real Hilbert space and ,
Facts & Assumptions
Given: A real Hilbert space and vectors .
A real Hilbert inner product is symmetric and bilinear, and (Hilbert space).
Proof
Expand the last square using [F1]: .
Substituting this expression into the right-hand side gives by bilinearity. This is the stated identity.
Depends on
Used by
- Backward Euler MINI step existence and discrete energy bound Theorem
- Backward Euler stability, first-order error for exact initial data, and reconstruction bounds Theorem
- Conditional monotone finite-element convergence Theorem
- Nonlinear backward Euler Galerkin existence and energy bounds Theorem
- Second-order Crank–Nicolson nodal error for exact initial data under H2 time regularity Theorem
- Semidiscrete Ritz error splitting and energy estimate Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Christian Clason, Introduction to Finite Element Methods, lecture notes (standard reference, not scraped)
- Long Chen, Finite Element Methods for Parabolic Equations (standard reference, not scraped)