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This article presents a stable numerical solution to the steady flow of thermodynamic compatible third grade fluid past a porous plate. Problem formulation is completed through partial slip condition. The inability of classical Lagrange-Galerkin methods to provide a stable approximate solution to the envisaged flow problem is demystified. To prevail over the instability issues, a streamline-upwind-Petrov-Galerkin method is invoked. The results thus substantiate an apposite agreement with theory. The stability of the approximate solution is testified and appropriate conclusions on flow profile are delineated. 2010 MSC: 76D15, 35C15, 76A05, Keywords: Third grade fluid, Partial slip, Streamline-upwind-Petrov-Galerkin, Finite element method