The paper concerns the three-dimensional Cauchy-Riemann type equations \(div u=\rho\), \(rot u=\zeta\) in \(D\subset {\mathbb{R}}^ 3\), \(u\cdot n=0\) on \(\partial D\), where D is a bounded domain, n the outward normal of \(\partial D\) and \(\rho\),\(\zeta\) are prescribed functions satisfying certain compatibility conditions. The numerical solution is studied from both the finite element and the finite difference points of view; both approaches rest on very similar foundations.
An integral identity is used to develop a finite element scheme that does not impose any restrictions on the finite element space. Convergence estimates are proofed using common techniques (for linear elements: O(h) in the \(H^ 1\)-norm; \(O(h^ 2)\) in the \(L^ 2\)-norm, Nitsche trick). As finite difference scheme a Keller box-scheme is used based on a least squares summation formulation. The scheme is not restricted to uniform grids; second-order-accuracy is proofed.