An efficient implicit spectral element method for time-dependent nonlinear diffusion equations by evaluating integrals at one quadrature point

We present an implicit spectral element method to approximate the solution of time-dependent nonlinear diffusion equations in complex geometries. We propose a nodal expansion to approximate derivatives of an unknown function, so integrals involving nonlinearities are evaluated at one quadrature point and discrete form of nonlinear diffusion equations is obtained without forming local and global stiffness matrices. Using the method-of-lines (MoL), the nonlinear partial differential equation is reduced to a nonlinear system of ordinary differential equations such that the MoL formulation is derived in terms of the differentiation matrix instead of the stiffness matrix. Since the arising system is stiff, an implicit method is employed to solve it. Also, with a simple algorithm we obtain a closed analytical form for the Jacobian matrix of the system of nonlinear algebraic equations. In addition, the differentiation matrices for triangular elements and deformed quadrilateral and hexahedral elements are obtained. For deformed elements an isoparametric mapping is used where the expansion coefficients are obtained with the aid of the Gordon-Hall mapping. Several benchmarks, including the scalar and system of these equations, are carried out. As physical applications, the cell motion model over long time, the tumour angiogenesis model on a mixed quadrangle-triangle mesh, and two and three-dimensional p -Laplace equations for various choices of the parameter p are considered.