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We present and analyze a first order least squares method for convectiondominated diffusion problems, which provides robust L2 a priori error estimatefor the scalar variable even if the given data f in L2 space. The noveltheoretical approach is to rewrite the method in the framework of discontinuousPetrov - Galerkin (DPG) method, and then show numerical stability by using akey equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014),pp. 537-552]. This new approach gives an alternative way to do numericalanalysis for least squares methods for a large class of differential equations.We also show that the condition number of the global matrix is independent ofthe diffusion coefficient. A key feature of the method is that there is nostabilization parameter chosen empirically. In addition, Dirichlet boundarycondition is weakly imposed. Numerical experiments verify our theoreticalresults and, in particular, show our way of weakly imposing Dirichlet boundarycondition is essential to the design of least squares methods - numericalsolutions on subdomains away from interior layers or boundary layers haveremarkable accuracy even on coarse meshes, which are unstructuredquasi-uniform.

First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems