Copeland, Dylan Matthew (0001-05). Negative-norm least-squares methods for axisymmetric Maxwell equations. Doctoral Dissertation. Thesis uri icon

abstract

  • We develop negative-norm least-squares methods to solve the three-dimensional Maxwell equations for static and time-harmonic electromagnetic fields in the case of axial symmetry. The methods compute solutions in a two-dimensional cross section of the domain, thereby reducing the dimension of the problem from three to two. To achieve this dimension reduction, we work with weighted spaces in cylindrical coordinates. In this setting, approximation spaces consisting of low order finite element functions and bubble functions are analyzed. In contrast to other methods for axisymmetric Maxwell equations, our leastsquares methods allow for discontinuous coefficients with large jumps and non-convex, irregular polygonal domains discretized by unstructured meshes. The resulting linear systems are of modest size, are symmetric positive definite, and can be solved very efficiently. Computations demonstrate the robustness of the methods with respect to the coefficients and domain shape.
  • We develop negative-norm least-squares methods to solve the three-dimensional
    Maxwell equations for static and time-harmonic electromagnetic fields in the case of
    axial symmetry. The methods compute solutions in a two-dimensional cross section
    of the domain, thereby reducing the dimension of the problem from three to two. To
    achieve this dimension reduction, we work with weighted spaces in cylindrical coordinates.
    In this setting, approximation spaces consisting of low order finite element
    functions and bubble functions are analyzed.
    In contrast to other methods for axisymmetric Maxwell equations, our leastsquares
    methods allow for discontinuous coefficients with large jumps and non-convex,
    irregular polygonal domains discretized by unstructured meshes. The resulting linear
    systems are of modest size, are symmetric positive definite, and can be solved very
    efficiently. Computations demonstrate the robustness of the methods with respect to
    the coefficients and domain shape.

publication date

  • May 2006