Extrapolation in the finite element method with penalty
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Consider the model problem Δu=f in Ω, u=0 on δΩ. Here Ω is a bounded open subset of Rn with smooth boundary, δΩ. The penalty method provides a method for obtaining an approximate solution without requiring the approximant to satisfy boundary conditions. Unfortunately, we pay a price for this convenience, namely loss of accuracy. We show that this difficulty may be alleviated by a particular type of extrapolation process. For a particular choice of boundary weight in the penalty method we show that repeated extrapolation always yields "optimal" error estimates in the energy norm.
KeywordsFinite Element Method Penalty Method Essential Boundary Condition Satisfy Boundary Condition Optimal Error Estimate
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