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## Hierarchical Bases

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Hierarchical scalar bases and hierarchical vector bases that span the Nédélec mixed-order spaces have been proposed for the most common two- and three-dimensional cells. These bases maintain appropriate continuity in a mesh containing multiple cell shapes. In particular, the vector bases maintain the tangential or normal continuity in the curl- or divergence-conforming case, respectively. The use of orthogonal scalar polynomials to systematically construct the basis functions is believed to offer a simpler approach for enhancing their linear independence as their polynomial order increases than the partial orthogonalization of the final vector functions. The process by which the basis functions were obtained is described in detail. Numerical results for the matrix condition numbers suggest that these hierarchical bases exhibit reasonable linear independence for large orders.

Chapter Contents:

• 5.1 The Ill-Conditioning Issue
• 5.2 Hierarchical Scalar Bases
• 5.2.1 Tetrahedral and Triangular Bases
• 5.2.2 Quadrilateral Bases
• 5.2.3 Brick Bases
• 5.2.4 Prism Bases
• 5.3 Hierarchical Curl-Conforming Vector Bases
• 5.3.1 Tetrahedral and Triangular Bases
• 5.3.2 Quadrilateral and Brick Bases
• 5.3.3 Prism Bases
• 5.3.4 Condition Number Comparison
• 5.4 Hierarchical Divergence-Conforming Vector Bases
• 5.4.1 Reference Variables on the Face in Common to Adjacent Cells
• 5.4.2 Tetrahedral Bases
• 5.4.3 Prism Bases
• 5.4.4 Brick Bases
• 5.4.5 Numerical Results and Comparisons with Other Bases
• 5.5 Conclusion
• References

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