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Trust me. QED

SIAM News

Heroux, Michael A.

Consider a standard SIAM journal article containing theoretical results. Each theorem has a proof that typically builds on previous developments. Since every theorem stems from a firm foundation, the research community can trust a result without further evidence. One could thus argue that a theorem does not require a proof because surely an author would not publish it if no proof existed to back it up. Furthermore, respectable reviewers and editors expect proofs without exception, and papers containing proof-less theorems will likely go unpublished.

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TYPE Journal Article YEAR 2019
OSTI

Compatible meshfree discretization of surface PDEs

Computational Particle Mechanics

Trask, Nathaniel A.; Kuberry, Paul

Meshfree discretization of surface partial differential equations is appealing, due to their ability to naturally adapt to deforming motion of the underlying manifold. In this work, we consider an existing scheme proposed by Liang et al. reinterpreted in the context of generalized moving least squares (GMLS), showing that existing numerical analysis from the GMLS literature applies to their scheme. With this interpretation, their approach may then be unified with recent work developing compatible meshfree discretizations for the div-grad problem in Rd. Informally, this is analogous to an extension of collocated finite differences to staggered finite difference methods, but in the manifold setting and with unstructured nodal data. In this way, we obtain a compatible meshfree discretization of elliptic problems on manifolds which is naturally stable for problems with material interfaces, without the need to introduce numerical dissipation or local enrichment near the interface. As a result, we provide convergence studies illustrating the high-order convergence and stability of the approach for manufactured solutions and for an adaptation of the classical five-strip benchmark to a cylindrical manifold.

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TYPE Journal Article YEAR 2019
DOIOSTI
Results 2151–2175 of 9,998
Results 2151–2175 of 9,998