This paper presents a general theory of maximally orthogonalized div-and curl-conforming higher order basis functions (HOBFs) over generalized wires, quadrilaterals, and hexahedra. In particular, all elements of such bases, necessary for fast and easy implementation, are listed up to order . Numerical results, given for div-conforming bases applied in an iterative method of moments solution of integral equations, show that the condition number and the number of iterations are a) much lower than in the case of other HOBFs of polynomial type and b) practically not dependent on the applied expansion order.Index Terms-Basis functions, higher order modeling, integral equations, method of moments (MoM), finite-element method (FEM), numerical techniques.

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Basic theory of matrix equilibration is presented, relating it to other techniques for decreasing the condition number of matrix equations obtained by the method of moments (MOM) applied to surface integral equations (SIEs). It is shown that matrix equilibration is a general technique that can be used for both (1) balancing field and source quantities in SIEs, which is used to decrease the condition number in the case of SIEs of mixed type and high contrast in material properties, and (2) scaling basis and test functions in MOM, which is used to decrease the condition number in the case of higher-order bases and patches of different sizes. In particular, it is demonstrated that a combination of such balancing and scaling can be performed using simple matrix equilibration based on magnitudes of diagonal elements and 2-norms of rows/columns of the MOM matrix.

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