A finite element procedure for approximately solving a linear elliptic boundary value problem is defined and analyzed. In this procedure $Lu = f$ is approximately solved in each element by requiring that a projection of the residual vanish and the local solutions are tied together by penalty equations which weakly impose the boundary conditions and the necessary degree of inter-element continuity. Optimal L² and H¹ error estimates are established.
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Percell et al. (1978) studied this question.
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