## A penalty correction method for elliptic problems and extrapolation with parabolic problems |

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### Contents

Preliminaries and notation | 7 |

Some a priori estimates | 11 |

The penalty correction method | 22 |

7 other sections not shown

### Common terms and phrases

advance one timestep analysis Applying Lemma approximating subspaces bounded Bramble completes the proof computationally compute constant C independent continuous penalty problem convergence estimates cp e H1 Define the sequence e e sh eigenfunctions eigenvalues elliptic problem exists a positive extends continuously extrapolation approximation f,cp finite element method fully discrete scheme g,cp gives Green's identity Hence high order approximation Hilbert space homeomorphic Hs(fi induction hypothesis isomorphism Lemma 4.1 holds linear combination matrix decomposition method of Section negative norms nonzero parabolic problems penalty approximation penalty correction method penalty method penalty solution polation positive constant positive definite positive real numbers positive semidefinite problem l.l Proof of Lemma prove Lemma quadratic forms Rz(T selfadjoint operators semidiscrete problem sequence of functions Sobolev space solution operator solution to l.l symmetric and positive Theorem 4.2 tion triangle inequality unique element Vandermonde matrix Wahlbin zero