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Stress singularities in fluid mechanics problems arise at points where there is an abrupt change in a boundary condition or in the boundary shape. In solving singular problems numerically, special attention is required around the singular point in order to achieve reasonable accuracy and convergence rates. The most common approach is to refine the grid around the singular point. However, this treatment cannot completely eliminate the numerical inaccuracies, e.g., spurious stress oscillations, which may contaminate the global solution. Some investigators modify the mathematical problem to alleviate the singularity by smoothing either the boundary or the boundary conditions. In this work, we acknowledge the singularity--incorporating the asymptotic solution into a finite element scheme to avoid inaccuracies due to the singularity. This idea has been successfully used in fracture mechanics with a variety of numerical methods, and it is extended here to solve Newtonian flow problems with finite elements. Two different approaches are followed for this purpose: (1) the singular element approach in which special elements that embody the radial form of the singularity are constructed around the singular point, and (2) the singular basis function approach in which the known local solution is subtracted from the governing equations. The stick-slip, the sudden-expansion, and the die-swell problems have been solved with the singular element method, and improved accuracy has been achieved with coarse meshes in the neighborhood of the singular point. In the die-swell problem, the convergence of the free surface is dramatically accelerated. A novel singular basis function method based on the subtraction of the exact asymptotic terms and on a double integration by parts is also proposed. When applied to st and ard Laplace-equation problems, this method improves the solution accuracy and gives more accurate singular coefficients than those obtained with other singular techniques. It also gives satisfactory results for the stick-slip problem. A comparison of the two methods is also made and their advantages and their limitations are discussed.