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In this paper, we develop and study algorithms for approximately solving linear algebraic systems: |${{\mathcal{A}}}_h^\alpha u_h = f_h$|⁠ , |$ 0< \alpha <1$|⁠ , for |$u_h, f_h \in V_h$| with |$V_h$| a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem |$ {{\mathcal{A}}}^\alpha u=f$| with |${{\mathcal{A}}}$|⁠ , for example, coming from a second-order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the method of Vabishchevich (2015, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys. , 282 , 289–302) that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval |$[0,1]$|⁠. Here we develop and study two time-stepping schemes based on diagonal Padé approximation to |$(1+x)^{-\alpha }$|⁠. The first one uses geometrically graded meshes in order to compensate for the singular behaviour of the solution for |$t$| close to |$0$|⁠. The second algorithm uses uniform time stepping, but requires smoothness of the data |$f_h$| in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of |$f_h$| playing a major role for the second method. Finally, we present numerical experiments for |${{\mathcal{A}}}_h$| coming from the finite element approximations of second-order elliptic boundary value problems in one and two spatial dimensions. [ABSTRACT FROM AUTHOR]

We present a negative-norm least-squares method for axisymmetric div–curl systems arising from Maxwell's equations for electrostatics and magnetostatics in three dimensions. The method approximates the solution in a two-dimensional meridian plane. To achieve this dimension reduction, we must work with weighted spaces in cylindrical co-ordinates. In this setting, a stable pair of approximation spaces is developed and analysed. We also report the results of some numerical experiments, which demonstrate a quasi-optimal convergence rate and a robustness with respect to the domain and coefficients. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic Maxwell equations. In particular, a "backslash" multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used. The method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator. This idea has previously been used in multigrid analysis of indefinite second order elliptic problems. However, the Maxwell application involves a nonelliptic indefinite operator. With the help of a few new estimates, the earlier ideas can still be applied. Some numerical experiments with lowest order Nedelec elements are also reported. [ABSTRACT FROM AUTHOR]