For high order finite elements (continuous piecewise polynomials) the conditioning of the basis is important. However, so far there seems no generally accepted concept of "a well-conditioned basis”, or a general strategy for how to obtain such representations. In this presentation, we use the $L^2$ condition number as a measure of the conditioning, and construct representations by frames such that the associated $L^2$ condition number is bounded independently of the polynomial degree. The main tools include the bubble transform, which is a stable decomposition of functions into local modes, and orthogonal polynomials on simplexes. We also include a brief discussion on potential applications in preconditioning. This is a joint work with Ragnar Winther.
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- Numerical Analysis Group Internal Seminar