### WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS

Kaibo Hu1, Ragnar Winther2

1. 1. School of Mathematics, University of Minnesota, 55455 Minneapolis, MN, USA;
2. Department of Mathematics, University of Oslo, 0316 Oslo, Norway
• Received:2018-04-27 Revised:2018-04-27 Published:2021-04-12
• Contact: Kaibo Hu,Email:khu@umn.edu
• Supported by:
The authors are grateful to Douglas N. Arnold, Richard S. Falk and Jinchao Xu for several discussions about the results of this paper. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 339643. The research of the first author leading to the results of this paper was partly carried out during his affiliation with the University of Oslo, partly supported by China Scholarship Council (CSC), project 201506010013.

Kaibo Hu, Ragnar Winther. WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS[J]. Journal of Computational Mathematics, 2021, 39(3): 333-357.

The purpose of this paper is to discuss representations of high order C0 finite element spaces on simplicial meshes in any dimension. When computing with high order piecewise polynomials the conditioning of the basis is likely to be important. The main result of this paper is a construction of representations by frames such that the associated L2 condition number is bounded independently of the polynomial degree. To our knowledge, such a representation has not been presented earlier. The main tools we will use for the construction is the bubble transform, introduced previously in[1], and properties of Jacobi polynomials on simplexes in higher dimensions. We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.

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