Element‐by‐element linear and nonlinear solution schemes

The preconditionded conjugate gradient method for solving a linear algebraic system of equations is recast to a form that permits sequential element‐by‐element calculations suitable for computations with finite element methods. This strategy has been implemented for solving the linear systems arising in a finite element approximation for the standard test example of Laplace's equation. The element‐by‐element strategy has also been applied to the sequence of linear systems obtained using a successive approximation scheme for a representative class of nonlinear problems. Little storage is needed for these schemes, and test computations have been made on microprocessor, minicomputer, main‐frame computers and special processors. The approach also appears appealing for calculations on parallel processors since individual element computations can be done in parallel.