Finite element patch approximations and alternating-direction methods

Numerical results using two modified Crank-Nicolson time discretizations are presented for finite element approximations to the nonlinear parabolic equation in 1Rd, c(x,u) − (aij (x,u)u,j),i + bi (x,u)u,i = f(x,t,u), where the summation convection on repeated indices is assumed. Both procedures use a local approximation to the coefficients which is based on patches of finite elements. With the first method, the coefficients are updated at each time step; however, only one matrix decomposition is required per problem. This method can exploit efficient direct methods for solving the resulting matrix problem. The second method is an alternating-direction variation which is valid for certain nonrectangular regions. With the alternating-direction method the resulting matrix problem can be solved as a series of one-dimensional problems, which results in a significant savings of time and storage over traditional techniques.