Uniform convergence of Galerkin’s method for splines on highly nonuniform meshes
Frank Natterer · Mathematics of Computation · 1977
Different sets of conditions for an estimate of the form \[ ‖ y − y π ‖ L ∞ ( a , b ) ⩽ C max i h i r + 1 ‖ y ( r + 1 ) ‖ L ∞ ( I i ) {\left \| {y - {y^\pi }} \right \|_{{L_\infty }(a,b)}} \leqslant C\max \limits _i h_i^{r + 1}{\left \| {{y^{(r + 1)}}} \right \|_{{L_\infty }({I_i})}} \] to hold are given. Here, y π {y^\pi } is the Galerkin approximation to the solution y of a boundary value problem for an ordinary differential equation, the trial functions being polynomials of degree ⩽ r \leqslant r on the subintervals I i = [ x i , x i + 1 ]