Analytical-numerical method of solving Poisson equation in complicated domains (Q1363266)

This work is devoted to an analytical-numerical method of solving the Dirichlet problem for the Poisson equation \(\Delta u=f\) with arbitrary right-hand side belonging to \(L_p(g)\), \(p>1\), in domains \(g\) having curvilinear boundary \(\partial g=\gamma \cup\Gamma\), when on the curve \(\Gamma\) the boundary function \(\varphi\) from \(L_2\) is defined, and a uniform boundary condition is set on \(\gamma\). This method is oriented to effectively determine not only the solution itself, but its derivatives in \(g\) up to the curve \(\gamma\), which may contain geometrical singularities. The essence of this method is to determine an explicit algorithm of constructing weight eigenfunctions \(v_n(w)\) of the Laplace operator.