The Ritz method minimizes a quadratic energy over a finite-dimensional trial space. For basis functions and bilinear form , its linear system is with and .
A symmetric coercive bilinear form defines the energy norm . In this norm the Ritz method is an orthogonal projection onto its finite-dimensional trial space.
If solves a variational problem and is its conforming Galerkin method approximation in , thenfor every .
For a continuous coercive bilinear form with continuity constant and coercivity constant , a conforming finite-element solution satisfiesFor a symmetric problem measured in its energy norm, the constant is one.
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The Ritz method is a variational technique used in mathematical analysis, particularly in the fields of applied mathematics and engineering, to find approximate solutions to complex problems, typically involving differential equations. It is particularly useful for problems in structural mechanics, quantum mechanics, and other fields where the governing equations can be difficult to solve exactly.