Gantumur Tsogtgerel | McGill University (original) (raw)

Uploads

Papers by Gantumur Tsogtgerel

In this article we initiate a systematic study of the well-posedness theory of the Einstein const... more In this article we initiate a systematic study of the well-posedness theory of the Einstein constraint equations on compact manifolds with boundary. This is an important problem in general relativity, and it is particularly important in numerical relativity, as it arises in models of Cauchy surfaces containing asymptotically flat ends and/or trapped surfaces. Moreover, a number of technical obstacles that appear when developing the solution theory for open, asymptotically Euclidean manifolds have analogues on compact manifolds with boundary. As a first step, here we restrict ourselves to the Lichnerowicz equation, also called the Hamiltonian constraint equation, which is the main source of nonlinearity in the constraint system. The focus is on low regularity data and on the interaction between different types of boundary conditions, which have has not been carefully analyzed before. In order to develop a well-posedness theory that mirrors the existing theory for the case of closed manifolds, we first generalize the Yamabe classification to nonsmooth metrics on compact manifolds with boundary. We then extend a result on conformal invariance to manifolds with boundary, and prove a uniqueness theorem. Finally, by using the method of suband super-solutions (order-preserving map iteration), we then establish several existence results for a large class of problems covering a broad parameter regime, which includes most of the cases relevant in practice.

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.