On Optimal Numerical Solution of Partial Differential Equations
Hans F. Weinberger · SIAM Journal on Numerical Analysis · 1972
The numerical solution of a linear initial or boundary value problem is formulated in the following manner. Given a discretization N from the linear space B of data to an n-space $E_n$ and an interpolation M from $E_m$ to the space $\Sigma $ of solutions, find a computation matrix Q so that the mapping $MQN$ approximates the solution operator S. It is shown that if B and $\Sigma $ are Banach spaces, then there exists at least one Q which is optimal in the sense that $||S - MQN||$ is minimized. When B and $\Sigma $ are Hilbert spaces, this minimum error can be obtained in terms of two simpler extremal problems which allows one to give upper and lower bounds for it. Examples involving the Laplace equation and the heat equation are presented.