Abstract
This chapter discusses variational finite element methods for initial value problems. In order of increasing generality one has finite element methods based on: (1) extremum principles, (2) stationary principles that are not extremum, (3) quasi-variational principles, and (4) the method of weighted residuals. The virtue of methods based on extremum principles is that one can develop numerical procedures that are guaranteed to be well-behaved in some sense. However, extremum principles exist only for restricted classes of equation. The advantage of the method of weighted residuals is that it can be applied to quite general equations, but it is not always clear how to choose the optimum weight factors. Stationary principles and quasi-variational principles help to choose suitable weighting functions. These considerations are to some extent independent of the fact that finite element methods usually involve weighting functions that are nonzero in only parts of the range of interest, which tends to produce stable numerical procedures.
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