Application of the multipole method to direct and inverse problems for the Grad–Shafranov equation with a nonlocal condition

Two homogeneous Dirichlet problems for the Grad–Shafranov equation with an affine right-hand side are considered in plane simply connected domains with a piecewise smooth boundary. The problems are denoted by and $(\mathrm{D})$, and $(\mathrm{A})$ the second involves a nonlocal condition. The corresponding inverse problems $(\mathrm{D}^{-1})$ and $(\mathrm{A}^{-1})$ of finding unknown parameters on the right-hand side of the equation from a given normal derivative of the solution to the respective direct problem are also considered. These problems arise in the computation of plasma flow characteristics in a tokamak. It is shown that the sought parameters can be found from two given quantities: (i) the normal derivative in the corresponding direct problem, which is physically interpreted as the magnetic field at an arbitrary single point $\tilde{x}$ of a special subset $\tilde{\Gamma}$ of the boundary $\Gamma$, and (ii) the integral of the normal derivative over $\Gamma$, which is physically interpreted as the total current through the tokamak cross section. Both problems are shown to be uniquely solvable, and necessary and sufficient conditions for unique solvability are given. A method for finding the desired parameters is proposed, which includes a technique for finding the subset $\tilde{\Gamma}$. The results are based, first, on the multipole method, which ensures the high-order accurate computation of the normal derivatives of the solution to direct problems $(\mathrm{D})$ and $(\mathrm{A})$, and, second, on the asymptotics of these derivatives as one of the parameters on the right-hand side of the equation tends to infinity.