Precise solution of Laplace’s equation

Abstract:

An approximate method is described for solving Laplace’s equation \[ \left \{ {\begin {array}{*{20}{c}} {\Delta u = 0} \\ {u{|_{\partial \Omega }} = g} \\ \end {array} \quad \begin {array}{*{20}{c}} {{\text {in}}\;\Omega = (0,1) \times (0,1),} \hfill \\ {{\text {on}}\;\partial \Omega } \hfill \\ \end {array} } \right .\] precisely in the sense of Aberth’s 1988 monograph. The algorithm uses singularity extraction, Fourier series methods, Taylor series methods, and interval analysis to construct an approximation $U(x,y)$ to the solution $u(x,y)$ at points in the square, and a uniform bound on the error $|U(x,y) - u(x,y)|$. The algorithm applies to problems in which the boundary data g is specified in terms of four elementary functions. The boundary data may be discontinuous at the corners.