Exact solution of an optimization problem generated by the three-dimensional Laplace equation

A one-parameter family of finite-dimensional spaces consisting of special three-dimensional splines of Lagrangian type is defined (the parameter $N$ is related to the dimension of the spline space). The solution of the boundary value problem for the Laplace equation given in a three-dimensional parallelepiped admits a representation in the form of a sum of four summands: a function linear in each of the three variables, and solutions of three particular boundary value problems generated by the original equation. In turn, these problems give rise to three problems of minimizing the functionals of residuals given in the indicated spline spaces. This decomposition allows one to study only one of the three optimization problems (the other two are symmetric in nature). A system of linear algebraic equations is obtained with respect to the coefficients of the optimal spline that gives the smallest discrepancy. It is shown that the system has a unique solution. The numerical solution of the system reduces to the implementation of the sweep method (the stability of this method holds). Numerical experiments show that with increasing $N,$ the minimum of the residual functional tends to zero.