The author gives a reformulation of the Dirichlet problem which, for the particular case of the Laplace operator in a domain \( \Omega\) takes the form: \( -\Delta u=f \) in \( \Omega\) and \( u=g \) on \( \Gamma=\partial\Omega \) in which \( f\in L^2(\Omega), g\in H^{1/2}(\Gamma)\). The Lagrange variational form of the Dirichlet problem is expressed as \( \int_{\Omega} \nabla u\cdot \nabla v-\int_\Gamma \lambda v=\int_\Omega fv \) and \(\int_\Gamma \mu v=\int_\Gamma \mu g \) and the problem is to find \( u\in H^1(\Omega)\), \( \lambda\in H^{-1/2}(\Gamma) \) such that the identities hold for all \( v\in H^1(\Omega)\) and \( \mu\in H^{-1/2}(\Gamma) \). But the natural bilinear form \( a(u,\lambda;v,\mu)= \int_{\Omega} \nabla u\cdot \nabla v-\int_\Gamma \lambda v +int_\Gamma \mu u\) is not coercive. Thus one seeks an equivalent coercive formulation of the problem in order to make existence and uniqueness transparent. For this purpose one adds to \( a\) the terms \( (u,v)_{1/2}, \) together with a multiple of \( (\lambda-\frac{\partial u}{\partial\nu}, \mu-\frac{\partial v}{\partial\nu}) \) and a term \( \sigma(\Delta u,\Delta v)\) to result in a stabilized/coercive form, and one then solves a corresponding stabilized variational problem. Here \( \frac{\partial}{\partial\mu}\) denotes the normal derivative, and \( \sigma\) is a bilinear form expressed in terms of wavelets. The stabilized variational problem then has the same solution as the original but with added advantages, based on hypotheses on the wavelets. The wavelets are assumed to form a basis for functions on \(\Gamma\) such that the Sobolev spaces in question have norms that can be expressed in terms of the moduli of the wavelet coefficients. Furthermore it is assumed that these wavelets can be lifted to obtain basis in \( \Omega\) with similar properties. These are nontrivial hypothesis that depend on the geometry of \( \Omega\) but the author gives some justification of why it is reasonable to assume the existence of such wavelets. The real benefit of the multiscale structure is that one can form uniform approximations to the solutions of the Dirichlet problem by working in the wavelet spaces up to a given scale. The discretized versions of the stabilized bilinear forms turn out to be uniformly coercive in a suitable sense. This is interpreted as meaning that accuracy of solutions can increase with scale while condition numbers of discretized problems remain bounded. These issues are addressed at least in abstract terms.