Constructive Approximation of Solutions to Linear Elliptic Boundary Value Problems

Let u be the solution of $Lu = \Delta u + au_x + bu_y + cu = 0$ on D, $u = f$ on $\partial D$, where f is continuous, the coefficients $a,b$ and c are analytic on D and c is nonpositive there. In Part I of this paper a set of particular solutions of $Lu = 0$ is constructed from the coefficients $a,b$ and c. These functions are independent of the domain D and represent generalizations of the harmonic polynomials $\operatorname{Re} (x + iy)^k $ and $\operatorname{Im} (x + iy)^k $ for $\Delta u = 0$. Special attention is paid to domains with corners in their boundaries and another family of particular solutions of $Lu = 0$ is constructed with the proper asymptotic behavior at these corners. It is shown that the solution, u, of the boundary value problem for $Lu = 0$ may be approximated arbitrarily well using finite linear combinations of these particular solutions. The norm used is the maximum norm over the closure of the domain D. Part II shows how to obtain such approximating linear combinations in practice. The technique is quite simple: choose M points on $\partial D$ and find a linear combination of N of these particular solutions, $N \leqq M$, which best approximates the given boundary values f at these M points. Obtaining such a linear combination is a linear programming problem and may be solved in practice. It is shown that as N and M go to infinity the computed linear combinations approach the solution of the problem uniformly. A posteriors error bounds are given for these approximate solutions of the boundary value problem. Finally, numerical results are presented for the special case where $a,b$ and c are polynomials in xand y.