Galerkin approximation with quadrature for the screen problem in \(\mathbb{R}^3\)
A semi-discrete Galerkin method for solving the single-layer equation \[ Vu(x)= {1\over 4\pi} \int_\Gamma {u(y)dy\over| x-y|}= f(x),\quad x\in\Gamma, \] where \(\Gamma\subset \mathbb{R}^2\) is a bounded region which is the union of a finite number of rectangles, with all the rectangle edges either parallel or perpendicular to each other, is investigated. Let \(\Gamma= \bigcup\Gamma_k\) be a finite partition of \(\Gamma\) into pairwise disjoint rectangles \(\Gamma_k\). The trial space \(S_h\) is the space of piecewise constant functions on the partition \(\Gamma= \bigcup\Gamma_k\). The Galerkin approximation \(u^G_k\in S_h\) is then defined as the solution of \[ (Vu^G_n, \xi)= (f,\xi),\quad\xi\in S_n.\tag{1} \] A semi-discrete scheme is obtained by approximating the \(L^2(\Gamma)\) inner product in (1) by special quadrature rules. Under appropriate conditions the stability of the semi-discrete scheme is proved. It is shown that the semi-discrete Galerkin approximations and the corresponding Galerkin approximations have convergence at the same rate in some norms.
scientific article; zbMATH DE number 1185925
scientific article