The Marker-and-Cell technique is modified and generalized to the flow problems of non-Newtonian fluid, especially of viscoelastic fluid. The solution can be obtained for reasonably high elastic Deborah number. Discussions are also on the bubble formation and growth at the orifice submerged under the Maxwell liquid in finite extent, which is a typical problem incorporating varieties of complexity, including the non-sphericity of the bubble, the viscoelasticity of the liquid and the complex flow field.