A finite-element method for computing nonlinear force-free fields
Motivated by a proposed iterative scheme for finding solution pairs \((\lambda,{\mathbf B})\) of \[ \operatorname {curl}{\mathbf B}=\lambda{\mathbf B}, \qquad \operatorname {div}{\mathbf B}= 0,\tag{1} \] subject to suitable boundary conditions on the boundary of a bounded, simply connected domain \(\Omega\), the authors are led to consider a linear boundary value problem associated with the transport equation \[ {\mathbf j}_0\cdot \operatorname {grad}\lambda+ \varepsilon\lambda= 0\tag{2} \] and another boundary value problem associated with the elliptic system \[ \operatorname {curl}{\mathbf B}-\operatorname {grad}p= {\mathbf j}_1, \qquad \operatorname {div}{\mathbf B}= 0,\tag{3} \] where \({\mathbf j}_0,{\mathbf j}_1\) are suitably given, \(\varepsilon\in ]0,\infty[\). After adding in a diffusion term \[ -h^r \operatorname {div}({\mathbf j}_0({\mathbf j}_0\cdot \operatorname {grad}) \lambda), \] \(r,h\in ]0,\infty[\) (as suggested by the streamline diffusion method) in the transport equation, both problems become tractible by variational methods. A corresponding finite-element method is considered, error estimates are found and numerical tests are discussed for both problems. Although, convergence of the initially suggested iterative scheme is still an open question, its applicability is finally numerically tested for a particular, explicit solution of the nonlinear system (1).
scientific article; zbMATH DE number 1755403