A Discontinuous Galerkin Method For Solving Fourth Order Curl Problem

The fourth order curl problem is of importance in the practical applications.For instance, it can describe the interaction of electrically conducting fluids with magnetic field at small length and short time scales.At present,there is little research on the basic theory,which conclude the well-posedness of the corresponding variational problem and regularity of weak solution,and numerical computation of the problem.We firstly provide the background and deduction of the problem on a 3-dimensional bounded Lipschitz polyhedron.Secondly,we bring forward the variational problem of the fourth order curl problem by introducing proper trial and testing function spaces,and prove that these trial and testing function spaces are Hilbert spaces and that the variational problem is well-posed.Furthermore,we obtain some regularity results of the weak solution by making use of the embedding relation between vector Sobolev spaces.Finally,based on the Nédélec finite element space,a discontinuous Galerkin(DG) method for solving the fourth order curl problem is presented,the related basic theory and an optimal error estimate are obtained.