In this paper, the discontinuous Petrov--Galerkin approximation of the Laplace eigenvalue problem is discussed. We consider in particular the primal and ultra weak formulations of the problem and prove the convergence together with a priori error estimates. Moreover, we propose two possible error estimators and perform the corresponding a posteriori error analysis. The theoretical results are confirmed numerically and it is shown that the error estimators can be used to design an optimally convergent adaptive scheme.

翻译：本文讨论了Laplace egenvalu 问题的不连续性Petrov-Galerkin近似值。我们特别考虑了问题的原始和极端弱的配方,并证明这与先验误差估计一致。此外,我们提出了两个可能的误差估计器,并进行了相应的事后误差分析。理论结果在数字上得到了确认,并表明误差估计器可用于设计一个最佳一致的适应方案。