On the accuracy of least squares methods in the presence of corner singularities.

*(English)*Zbl 0573.65081This paper is concerned with optimal convergence results for least squares methods in the presence of corner singularities. Such optimality is obtained by giving an alternate least squares approximation in weighted Sobolev spaces. The key feature of this paper is that the error estimates are in unweighted norms like \(L^ 2\). Throughout the paper, the authors consider the least squares approximation of the problem
\[
(1)\quad \Delta \phi +q\phi =f\quad in\quad \Omega,\quad \phi =0\quad on\quad \partial \Omega.
\]
First they give a corresponding weighted variational formulation of this problem. Next, they recall a regularity result concerning problems set on a domain with a corner, and they develop some approximation results. The analysis of error estimates has a structure similar to that found in the analysis of mixed method. In particular, the error estimates concern: \(\epsilon =\phi -\phi_ h\) and \(e=u-u_ h\) where \(\phi\) is the solution of equation (1), \(u=\text{grad} \phi\), \(\phi_ h\in S_ h\subseteq \overset\circ H^ 1(\Omega)\) and \(u_ h\in \vec V_ h\subseteq \vec W^{1,\beta}(\Omega)\) are the corresponding solutions of the weighted variational formulation. Then, the main results are (theorems 4.1 and 4.2):
\[
\| \phi -\phi_ h\|_ 0\leq Ch^ 2\{\| f\|_{0,\beta}+\| f\|_{1,2\beta}\},
\]
\[
\| u-u_ h\|_ 0\leq Ch^ 2\{\| f\|_{0,\beta}+\| f\|_{1,2\beta}\}+C\inf_{\hat u_ h\in \vec V_ h}\| u-\hat u_ h\|_ 0.
\]
Finally some convincing numerical results are discussed: i) on a L-shaped membrane with re- entrant corner \(\theta =3\pi /2\); ii) on a square region with a crack.

Reviewer: M.Bernadou

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

##### Keywords:

finite element methods; weighted norms; Helmholtz equation; optimal convergence; least squares methods; corner singularities; weighted Sobolev spaces; error estimates; membrane; crack
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\textit{C. L. Cox} and \textit{G. J. Fix}, Comput. Math. Appl. 10, 463--475 (1984; Zbl 0573.65081)

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##### References:

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