Singular Perturbation Method for Smart Building Temperature Control Using Occupant Feedback: Singular Perturbation Method For Building Temperature Control

Gupta, Santosh K.; Kar, Koushik; Mishra, Sandipan; Wen, John T.

doi:10.1002/asjc.1587

We propose and investigate the application of alternative enriched test spaces in the discontinuous Petrov–Galerkin (DPG) finite element framework for singular perturbation linear problems, with an emphasis on 2D convection-dominated diffusion. Providing robust L2 error estimates for the field variables is considered a convenient feature for this class of problems, since this normwould not account for the large gradients present in boundary layers. With this requirement in mind, Demkowicz and others have previously formulated special test norms, which through DPG deliver the desired L2 convergence. However, robustness has only been verified through numerical experiments for tailored test normswhich are problem-specific,whereas the quasi-optimal test norm (not problem specific) has failed such tests due to the difficulty to resolve the optimal test functions sought in the DPG technology. To address this issue (i.e. improve optimal test functions resolution for the quasi-optimal test norm), we propose to discretize the local test spaces with functions that depend on the perturbation parameter ϵ. Explicitly,wework with B-spline spaces defined on an ϵ-dependent Shishkin submesh. Two examples are run using adaptive h-refinement to compare the performance of proposed test spaces with that of standard test spaces. We also include a modified norm and a continuation strategy aiming to improve time performance and briefly experiment with these ideas.

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