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Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems. (English) Zbl 1314.76034

Computing 95, No. 5, 425-448 (2013); erratum ibid. 96, No. 11, 1111-1112 (2014).
Summary: This paper is concerned with the extension of the algebraic flux-correction (AFC) approach to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming \(P_1\) and \(Q_1\) finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix-Raviart element [M. Crouzeix and P. A. Raviart, Rev. Franc. Automat. Inform. Rech. Operat., R 7, No. 3, 33–76 (1974; Zbl 0302.65087)] on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher-Turek elements [R. Rannacher and S. Turek, Numer. Methods Partial Differ. Equations 8, No. 2, 97–111 (1992; Zbl 0742.76051)]. The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection-diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix-Raviart element. Good resolution of smooth and discontinuous profiles is attested to \(Q_1^{\mathrm{nc}}\) approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix-vector multiplications is addressed.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
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