Journal
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
Volume 32, Issue 2, Pages 479-509Publisher
WILEY
DOI: 10.1002/num.22002
Keywords
adaptive methods; a posteriori estimates; approximation theory; a priori estimates; contraction; convergence; duality; elliptic equations; goal oriented; nonsymmetric problems; optimality; quasi-orthogonality; residual-based error estimator
Categories
Funding
- NSF [1065972, 1217175, 1262982]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1318480, 1262982, 1217175, 1065972] Funding Source: National Science Foundation
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In this article, we develop convergence theory for a class of goal-oriented adaptive finite element algorithms for second-order nonsymmetric linear elliptic equations. In particular, we establish contraction results for a method of this type for Dirichlet problems involving the elliptic operator Lu = del.(A del u) - b . del u - cu, with A Lipschitz, symmetric positive definite, with b divergence-free, and with c >= 0. We first describe the problem class and review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We then describe a goal-oriented variation of standard AFEM. Following the recent work of Mommer and Stevenson for symmetric problems, we establish contraction and convergence of the goal-oriented method in the sense of the goal function. Our analysis approach is signficantly different from that of Mommer and Stevenson, combining the recent contraction frameworks developed by Cascon, Kreuzer, Nochetto, and Siebert; by Nochetto, Siebert, and Veeser; and by Holst, Tsogtgerel, and Zhu. We include numerical results, demonstrating performance of our method with standard goal-oriented strategies on a convection problem. (C) 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 479-509, 2016
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