On the use of slope limiters for the design of recovery based error indicators

Gradient recovery techniques for the design of a posteriori error indicators are reviewed in the context of fluid dynamic problems featuring shocks and discontinuities. An edgewise slope limiting approach 24 tailored to linear finite element discretizations is presented. The improved gradient values at edge midpoints are recovered as the limited average of constant slopes from adjacent triangles. Furthermore, the low-order gradients may serve as natural upper and lower bounds to be imposed on the edge slopes. To this end, approved techniques such as averaging projection 33 , the (superconvergent) Zienkiewicz-Zhu patch recovery (SPR 34 ) and polynomial preserving recovery (PPR 29 ) are used to predict high-order gradients. A slope limiter is applied edge-by-edge to correct the provisional edge gradient values subject to geometric constraints. In either case, a second order accurate quadrature rule is employed to measure the difference between consistent and reconstructed slopes in the (local) L2-norm which provides a usable indicator for grid adaptation. The algebraic flux correction (AFC) methodology 14-19 is equipped with adaptive mesh refinement/coarsening procedures governed by the recovery based error indicator. The adaptive algorithm is applied to inviscid compressible flows at high Mach numbers.

[1]  Rainald Löhner.  Robust, Vectorized Search Algorithms for Interpolation on Unstructured Grids , 1995 .

[2]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE , 1995 .

[3]  A. Jameson.  Computational algorithms for aerodynamic analysis and design , 1993 .

[4]  R. Bank,et al.  Some Refinement Algorithms And Data Structures For Regular Local Mesh Refinement , 1983 .

[5]  Andre Garon,et al.  A robust implementation of Zienkiewicz and Zhu's local patch recovery method , 1995 .

[6]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[7]  María Cecilia Rivara,et al.  Design and data structure of fully adaptive, multigrid, finite-element software , 1984, ACM Trans. Math. Softw..

[8]  Zhimin ZhangDepartment,et al.  Analysis of the Superconvergent Patch RecoveryTechnique and A Posteriori Error Estimator in theFinite Element Method ( II ) , 1995 .

[9]  Stefan Turek,et al.  Flux correction tools for finite elements , 2002 .

[10]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[11]  R. LeVeque.  Scalar Conservation Laws , 1992 .

[12]  Dmitri Kuzmin,et al.  Algebraic Flux Correction I. Scalar Conservation Laws , 2005 .

[13]  Zhimin Zhang,et al.  Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II) , 1998 .

[14]  Richard Abraham Shapiro.  Adaptive Finite Element Solution Algorithm for the Euler Equations , 1991 .

[15]  W. Hackbusch.  Iterative Solution of Large Sparse Systems of Equations , 1993 .

[16]  A. Hannukainen,et al.  Computational Technologies for Reliable Control of Global and Local Errors for Linear Elliptic Type Boundary Value Problems , 2007 .

[17]  Zhimin Zhang.  Ultraconvergence of the patch recovery technique II , 2000, Math. Comput..

[18]  Dmitri Kuzmin,et al.  Algebraic Flux Correction II. Compressible Euler Equations , 2005 .

[19]  P. Hansbo.  A free-Lagrange finite element method using space-time elements , 2000 .

[20]  Dmitri Kuzmin,et al.  On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection , 2006, J. Comput. Phys..

[21]  Achi Brandt,et al.  Multilevel Adaptive Computations in Fluid Dynamics , 1980 .

[22]  Stefan Turek,et al.  Multidimensional FEM‐FCT schemes for arbitrary time stepping , 2003 .

[23]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[24]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[25]  O. Zienkiewicz,et al.  Analysis of the Zienkiewicz–Zhu a‐posteriori error estimator in the finite element method , 1989 .

[26]  Zhimin Zhang,et al.  A Posteriori Error Estimates Based on the Polynomial Preserving Recovery , 2004, SIAM J. Numer. Anal..

[27]  D. Kuzmin,et al.  A semi-implicit FEM-FCT algorithm for efficient treatment of time-dependent problems , 2005 .

[28]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[29]  Stefan Turek,et al.  Flux-corrected transport : principles, algorithms, and applications , 2005 .

[30]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[31]  Dmitri Kuzmin,et al.  Adaptive mesh refinement for high‐resolution finite element schemes , 2006 .