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## Representation of Vector Fields in Two and Three Dimensions Using Low-Degree Polynomials

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In this chapter, we consider the representation of vector functions (often referred to as “vector fields”) with low-order (constant and linear) polynomial basis functions on simple cells, such as triangles or quadrilateral cells in two dimensions or tetrahedrons and bricks in three dimensions. As will soon be apparent, there are multiple ways of defining vector basis functions, and therefore the approach requires some consideration. The proper representation of a function depends on what will be done with it-do we need to compute the curl of the function, for instance? If so, the representation might be different than if we need to compute the divergence of the function. We use the term curl conforming to denote the space of vector functions that maintain first-order tangential-vector continuity throughout the domain and can be differentiated via the curl operation, without producing unbounded or generalized functions (Dirac delta functions) in the process. The term divergence conforming is used to denote the complementary space of vector functions that maintain first-order normal-vector continuity throughout the domain and can therefore be differentiated via the divergence operation. (First order or C0 continuity is continuity of the function itself, but not necessarily continuity of its first derivatives.) The simple low-order polynomial vector basis functions in widespread use are either curl conforming or divergence conforming; seldom we will use functions that maintain complete continuity and belong to both the curl-conforming and divergence-conforming spaces, although it is possible to define such functions.

Chapter Contents:

• 3.1 Two-Dimensional Vector Functions on Triangles
• 3.1.1 Linear Curl-Conforming Vector Basis Functions
• 3.1.2 A Simpler Curl-Conforming Representation on Triangles
• 3.1.3 Alternate Approach: A Divergence-Conforming Representation on Triangles
• 3.2 Tangential-Vector versus Normal-Vector Continuity: Curl-Conforming and Divergence-Conforming Bases
• 3.3 Two-Dimensional Representations on Rectangular Cells
• 3.4 Quasi-Helmholtz Decomposition in 2D: Loop and Star Functions
• 3.5 Projecting between Curl-Conforming and Divergence-Conforming Bases
• 3.6 Three-Dimensional Representation on Tetrahedral Cells: Curl-Conforming Bases
• 3.7 Three-Dimensional Representation on Tetrahedral Cells: Divergence-Conforming Bases
• 3.8 Three-Dimensional Expansion on Brick Cells: Curl-Conforming Case
• 3.9 Divergence-Conforming Bases on Brick Cells
• 3.10 Quasi-Helmholtz Decomposition on Tetrahedral Meshes
• 3.11 Vector Basis Functions on Skewed Meshes or Meshes with Curved Cells
• 3.11.1 Base and Reciprocal Base Vectors
• 3.11.2 Covariant and Contravariant Projections
• 3.11.3 Derivatives in the Parent Space
• 3.11.4 Restriction to Surfaces
• 3.12 The Mixed-Order Nédélec Spaces
• 3.13 The De Rham Complex
• 3.14 Conclusion
• References

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