One-dimensional linear advection–diffusion equation: Analytical and finite element solutions

In this paper, a time dependent one-dimensional linear advection–diffusion equation with Dirichlet homogeneous boundary conditions and an initial sine function is solved analytically by separation of variables and numerically by the finite element method. It is observed that when the advection becomes dominant, the analytical solution becomes ill-behaved and harder to evaluate. Therefore another approach is designed where the solution is decomposed in a simple wave solution and a viscous perturbation. It is shown that an exponential layer builds up close to the downstream boundary. Discussion and comparison of both solutions are carried out extensively offering the numericist a new test model for the numerical integration of the Navier–Stokes equatio

A posteriori optimization of parameters in stabilized methods for convection-diffusion problems --- Part I

Stabilized finite element methods for convection-dominated problems require the choice of appropriate stabilization parameters. From numerical analysis, often only their asymptotic values are known. This paper presents a general framework for optimizing the stabilization parameters with respect to the minimization of a target functional. Exemplarily, this framework is applied to the SUPG finite element method and the minimization of a residual-based error estimator and error indicator. Benefits of the basic approach are shown and further improvements are discussed

Local projection stabilisation for convection-diffusion-reaction equations using a biorthogonal system and adaptive refinement

We consider a local projection stabilisation based on biorthogonal systems and adaptive refinement for convection-diffusion-reaction differential equations. The local projection stabilisation and adaptive finite element method are both based on a biorthogonal system. We investigate the numerical efficiency of the approach when compared to the standard finite element method. Numerical examples are presented to demonstrate the performance of the approach. References R. Becker and M. Braack. A finite element pressure gradient stabilization for the Stokes equations based on local projections. In:Calcolo 38.4 (2001), pp. 173–199. doi: 10.1007/s10092-001-8180-4 M. Braack and E. Burman. Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43.6 (2006), pp. 2544–2566. doi: 10.1137/050631227 D. Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, 2001. url: https://www.cambridge.org/au/academic/subjects/mathematics/numerical-analysis/finite-elements-theory-fast-solvers-and-applications-solid-mechanics-3rd-edition?format=PB S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer–Verlag, 1994. doi: 10.1007/978-0-387-75934-0 C. Carstensen, M. Feischl, M. Page, and D. Praetorius. Axioms of adaptivity. Comput. Math. Appl. 67.6 (2014), pp. 1195–1253. doi: 10.1016/j.camwa.2013.12.003 L. Chen, P. Sun, and J. Xu. Multilevel homotopic adaptive finite element methods for convection dominated problems. Domain Decomposition Methods in Science and Engineering. Ed. by T. J. Barth, M. Griebel, D. E. Keyes, R. M. Nieminen, D. Roose, T. Schlick, R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Widlund, and J. Xu. Springer, 2005, pp. 459–468. doi: 10.1007/3-540-26825-1_47 S. A. Funken and A. Schmidt. Adaptive mesh refinement in 2D—An efficient implementation in Matlab. Comput. Meth. Appl. Math. 20.3 (2020), pp. 459–479. doi: 10.1515/cmam-2018-0220 p. C131). D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, Springer-Verlag, 2001. doi: 10.1007/978-3-642-61798-0 V. John, P. Knobloch, and J. Novo. Finite elements for scalar convection-dominated equations and incompressible flow problems: A never ending story? Comput. Visual. Sci. 19.5 (2018), pp. 47–63. doi: 10.1007/s00791-018-0290-5 C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. Dover Books on Mathematics. Dover Publications, 2012. url: https://store.doverpublications.com/048646900x.html B. P. Lamichhane. Higher order mortar finite elements with dual Lagrange multiplier spaces and applications. PhD thesis. University of Stuttgart, 2006. doi: 10.18419/opus-4770 B. P. Lamichhane and J. A. Shaw-Carmody. A local projection stabilisation for convection-diffusion-reaction equations using biorthogonal systems. J. Comput. Appl. Math. 393, 113542 (2020). doi: 10.1016/j.cam.2021.113542 G. Matthies, P. Skrzypacz, and L. Tobiska. Stabilization of local projection type applied to convection-diffusion problems with mixed boundary conditions. Elec. Trans. Numer. Anal. 32 (2008), pp. 90–105. url: https://etna.math.kent.edu/volumes/2001-2010/vol32/abstract.php?vol=32&pages=90-105 H.-G. Roos, M. Stynes, and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems. Springer, 2008. doi: 10.1007/978-3-540-34467-4 P. Sun, L. Chen, and J. Xu. Numerical Studies of Adaptive Finite Element Methods for Two Dimensional Convection-Dominated Problems. J. Sci. Comput. 43 (2010), pp. 24–43. doi: 10.1007/s10915-009-9337-