Continuous interior penalty finite element methods for Sobolev equations with convection‐dominated term

We consider implicit and semi‐implicit time‐stepping methods for continuous interior penalty (CIP) finite element approximations of Sobolev equations with convection‐dominated term. Stability is obtained by adding an interior penalty term giving L² ‐control of the jump of the gradient over element faces. Several $cal {A}$ ‐stable time‐stepping methods are analyzed and shown to be unconditionally stable and optimally convergent. We show that the contribution from the gradient jumps leading to an extended matrix pattern may be extrapolated from previous time steps, and hence handled explicitly without loss of stability and accuracy. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012