2011/08/18 by Michael Holst, Holst, Michael, Sara Pollock +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1108.3660
openalex publication_date 2011/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we develop convergence theory for a class of goal-oriented\nadaptive finite element algorithms for second order nonsymmetric linear\nelliptic equations. In particular, we establish contraction results for a\nmethod of this type for Dirichlet problems involving the elliptic operator L u\n= div (A grad u) - (b,grad u) - cu, with A Lipschitz, almost-everywhere\nsymmetric positive definite, with b divergence-free, and with c >= 0. We first\ndescribe the problem class and review some standard facts concerning conforming\nfinite element discretization and error-estimate-driven adaptive finite element\nmethods (AFEM). We then describe a goal-oriented variation of standard AFEM\n(GOAFEM). Following the recent work of Mommer and Stevenson for symmetric\nproblems, we establish contraction of GOAFEM and convergence in the sense of\nthe goal function. Our analysis approach is signficantly different from that of\nMommer and Stevenson, combining the recent contraction frameworks developed by\nCascon, Kreuzer, Nochetto and Siebert; by Nochetto, Siebert and Veeser; and by\nHolst, Tsogtgerel and Zhu. We include numerical results demonstrating\nperformance of our method with standard goal-oriented strategies on a\nconvection problem.\n