p‐version least‐squares finite element formulation of Burgers' equation

A p‐version least‐squares finite element formulation for non‐linear problems is presented and applied to the steady‐state, one‐dimensional Burgers' equation. The second‐order equation is recast as a set of first‐order equations which permit the use of C0 elements. The primary and auxiliary variables are approximated using equal‐order p‐version hierarchical approximation functions. The system of non‐linear simultaneous algebraic equations resulting from the least‐squares process is solved using Newton's method with a line search. The use of ‘exact’ and ‘reduced’ quadrature rules is investigated and the results are compared. The formulation is found to produce excellent results when the ‘exact’ integration rule is used. The combination of least‐squares finite element formulation and p‐version works extremely well for Burgers' equation and appears to have great potential in fluid dynamics problems.