Performance of space-time coupled least-squares spectral element methods for parabolic problems

We study the performance of space-time coupled least-squares spectral element method (LSSEM) for parabolic initial boundary value problem (IBVP) for different values of element size h, the time step k, the degree q in the time variable and the degree p in each of the space variables. We divide the space domain into a number of shape regular quadrilaterals of size h and the time step k is proportional to h(2). Each shape regular quadrilateral will be mapped to the square (-1, 1) x (-1, 1). In each square the solution will be defined as a polynomial of degree p for the space variables and degree q for the time variable. For the p version of the method h remains fixed and p increases, for the the h-version of the method p remains fixed and h decreases but for the hp-version of the method h decreases and at the same time p also increases. In this method, k is proportional to h(2) (say k = ch(2)) and for the p-version of the method, q is proportional to p(2) (say q = c'p(2)). The performance of the method is discussed for different values of c and c'. The method we discussed here is spectral in both space and time and are non-conforming.