On exact solution of optimization task generated by the Laplace equation

A one-parameter family of finite-dimensional spaces consisting of special two-dimensional splines of Lagrangian type is defined (the parameter $N$ is related to the dimension of the space). The Laplace equation generates in each such space the problem of minimizing the residual functional. The existence and uniqueness of optimal splines are proved. For their coefficients and residuals, exact formulas are obtained. It is shown that with increasing $N,$ the minimum of the residual functional is ${\rm O}(N^{-5}),$ and the special sequence consisting of optimal splines is fundamental.