This paper presents a new computational strategy along with a computational and mathematical framework for computing non-weak numerical solutions of stationary and time dependent partial differential equations. This approach utilizes strong form of the governing differential equations (GDE) and least squares approach in constructing the integral form. This new proposed approach is applied to one dimensional transient gasdynamics equation in Eulerian frame of reference using ρ, u, T as dependent variables. The currently used finite element approaches seek convergence of a solution in a fixed order space by h, p, or hp-adaptive processes. The fundamental point of departure in the proposed approach is that we seek the convergence of the computed converged solution over the spaces of different orders containing the basis functions. With this approach, dramatically higher convergence rates than those obtained for h, p, or hp-processes are achievable and the sequence of progressively converged solutions over the spaces of progressively increasing order in fact converge to the strong solution (analytical or theoretical) of the partial differential equation. It is demonstrated using one-dimensional transient Navier-Stokes equations for compressible fluid flow in Eulerian frame of reference, that in the presence of physical diffusion and dissipation, our computed solutions have exactly the same characteristics as the strong solutions. Riemann shock tube is used as a model problem.