Abstract: In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferen... more Abstract: In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferential of the form as the triangulation is refined.
Abstract. The Cauchy-Kovalevskaya theorem, characteristic surfaces, and the notion of well posedn... more Abstract. The Cauchy-Kovalevskaya theorem, characteristic surfaces, and the notion of well posedness are discussed. We review some basic facts about analytic functions of a single variable in Section 1, which can be skipped. I thank Ibrahim for making his class notes available to me.
This paper presents adaptive boundary element methods for positive, negative, as well as zero ord... more This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.
Abstract: In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferen... more Abstract: In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferential of the form as the triangulation is refined.
Abstract. The Cauchy-Kovalevskaya theorem, characteristic surfaces, and the notion of well posedn... more Abstract. The Cauchy-Kovalevskaya theorem, characteristic surfaces, and the notion of well posedness are discussed. We review some basic facts about analytic functions of a single variable in Section 1, which can be skipped. I thank Ibrahim for making his class notes available to me.
This paper presents adaptive boundary element methods for positive, negative, as well as zero ord... more This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.
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