Finite element steady-state solutions of the traveling magnetic field problem

Abstract

The paper is concerned with stability and accuracy of n-order finite element (FE) steady-state solutions of traveling magnetic field problem. It is found that the odd-order FE solutions ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is the odd number) are stable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 &lt; \sigma|u|h|\nu &lt; f(n) \simeq 2.0 + 1.4(n - 1)</tex> , and that the even-order FE solutions ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is the even number) are unconditionally stable. The consistent domain is also proposed, in which the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -order FE solutions are stable and of 2 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -order accuracy. Moreover, three-dimensional cases are dealt with, and the comparison with upwind methods is given. The merit and limits of the n-order FE method are finally cleared.