Cubic spline approximation based on half-step discretization for 2D quasilinear elliptic equations

We report a new cubic spline approximation based on half-step discretization of order 2 in y- and order 4 in x-directions, for 2D quasi-linear elliptic PDEs. We use only two extra half-step points in x-direction and a central point. The cubic spline method is directly obtained from the continuity of first derivative terms and is applicable to elliptic problems irrespective of coordinates, which is the main advantage of our work. The error analysis of a model problem is discussed in details. Some benchmark problems are solved in order to test the numerical stability and accuracy of the method.