  In the paper, we construct a new quadratic spline-wavelet basis on the
interval and a unit square satisfying homogeneous Dirichlet boundary conditions
of the first order. Wavelets have one vanishing moment and the shortest support
among known quadratic spline wavelets adapted to the same type of boundary
conditions. Stiffness matrices arising from a discretization of the
second-order elliptic problems using the constructed wavelet basis have
uniformly bounded condition numbers and the condition numbers are small. We
present quantitative properties of the constructed basis. We provide numerical
examples to show that the Galerkin method and the adaptive wavelet method using
our wavelet basis requires smaller number of iterations than these methods with
other quadratic spline wavelet bases. Moreover, due to the short support of the
wavelets one iteration requires smaller number of floating point operations.
