Adaptive mesh refinement for high‐resolution finite element schemes

Journal of Computational Physics

Shashkov

Svyatsky

2009

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Nonlinear constrained finite element approximations to anisotropic diffusion problems are considered. Starting with a standard (linear or bilinear) Galerkin discretization, the entries of the stiffness matrix are adjusted so as to enforce sufficient conditions of the discrete maximum principle (DMP). An algebraic splitting is employed to separate the contributions of negative and positive off-diagonal coefficients which are associated with diffusive and antidiffusive numerical fluxes, respectively. In order to prevent the formation of spurious undershoots and overshoots, a symmetric slope limiter is designed for the antidiffusive part. The corresponding upper and lower bounds are defined using an estimate of the steepest gradient in terms of the maximum and minimum solution values at surrounding nodes. The recovery of nodal gradients is performed by means of a lumped-mass L 2 projection. The proposed slope limiting strategy preserves the consistency of the underlying discrete problem and the structure of the stiffness matrix (symmetry, zero row and column sums). A positivity-preserving defect correction scheme is devised for the nonlinear algebraic system to be solved. Numerical results and a grid convergence study are presented for a number of anisotropic diffusion problems in two space dimensions.

Section: Discrete Maximum Principlementioning

confidence: 99%

“…As already mentioned, the Galerkin approximation based on (16) and (19) gives rise to a symmetric stiffness matrix A = {a ij } with zero row and column sums…”

Section: Algebraic Splittingmentioning

confidence: 99%

A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems

Journal of Computational Physics

Shashkov

Svyatsky

2009

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“…For system (24) to have a unique solution, its rank has to be at least equal to the number of terms in the polynomial expansion (22). In the two dimensional case this implies that m ≥ (k + 1)(k + 2)/2.…”

Section: Limited Gradient Reconstructionmentioning

confidence: 99%

“…However, it is easy to construct meshes for which two boundary components are only separated by one layer of elements such that no interior patches are available 20 . Moreover, the choice of the interior patch Ω i may not be unique for unstructured triangulations 24 . Since the advent of the superconvergent patch recovery (SPR) technique 34 its super-and even ultraconvergence property has been analyzed extensively in the literature 28,31,32 .…”

Section: Limited Gradient Reconstructionmentioning

confidence: 99%

“…Recall that the amount of artificial dissipation that outlasts the flux limiter depends on the interplay of internodal fluxes which are proportional to the edgewise solution difference multiplied by some (anti-)diffusion coefficient. Therefore, it is quite expedient to bisect the edge with the largest solution variation or the largest antidiffusive flux 24 .…”

Section: Limited Gradient Reconstructionmentioning

confidence: 99%

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On the Use of Slope Limiters for the Design of Recovery Based Error Indicators

Möller

Numerical Mathematics and Advanced Applications

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Abstract. 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) L 2 -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.

Implicit finite element schemes for the stationary compressible Euler equations

Gurris

Numerical Methods in Fluids

Turek

2011

SUMMARY A semi‐implicit finite element scheme and a Newton‐like solver are developed for the stationary compressible Euler equations. Since the Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable, the troublesome antidiffusive part is constrained within the framework of algebraic flux correction. A generalization of total variation diminishing (TVD) schemes is employed to blend the original Galerkin scheme with its nonoscillatory low‐order counterpart. Unlike standard TVD limiters, the proposed limiting strategy is fully multidimensional and readily applicable to unstructured meshes. However, the nonlinearity and nondifferentiability of the limiter function makes efficient computation of stationary solutions a highly challenging task, especially in situations when the Mach number is large in some subdomains and small in other subdomains. In this paper, a semi‐implicit scheme is derived via a time‐lagged linearization of the Jacobian operator, and a Newton‐like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. A boundary Riemann solver is used to avoid unphysical boundary states. It is shown that the proposed approach offers unconditional stability, as well as higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. The overall spatial accuracy of the constrained scheme and the benefits of the new boundary treatment are illustrated by grid convergence studies for 2D benchmark problems. Copyright © 2011 John Wiley & Sons, Ltd.