Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes

The final publication is available at Springer via http://dx.doi.org/10.1007/s00211-016-0808-zFor the case of approximation of convection–diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions

Maximum-principle preserving space-time isogeometric analysis

In this work we propose a nonlinear stabilization technique for convection-diffusion-reaction and pure transport problems discretized with space-time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space-time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space-time discretization can become highly costly. Therefore, we also propose a partitioned space-time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection-diffusion problems in one and two dimensions

A unified analysis of Algebraic Flux Correction schemes for convection-diffusion equations

Recent results on the numerical analysis of Algebraic Flux Correction (AFC) finite element schemes for scalar convection-diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion scheme. Then, specific versions of the method, this is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme

Monotonicity-preserving finite element methods for hyperbolic problems

This thesis covers the development of monotonicity preserving finite element methods for hyperbolic problems. In particular, scalar convection-diffusion and Euler equations are used as model problems for the discussion in this dissertation. A novel artificial diffusion stabilization method has been proposed for scalar problems. This technique is proved to yield monotonic solutions, to be \ac{led}, Lipschitz continuous, and linearity preserving. These properties are satisfied in multiple dimensions and for general meshes. However, these results are limited to first order Lagrangian finite elements. A modification of this stabilization operator that is twice differentiable has been also proposed. With this regularized operator, nonlinear convergence is notably improved, while the stability properties remain unaltered (at least, in a weak sense). An extension of this stabilization method to high-order discretizations has also been proposed. In particular, arbitrary order space-time isogeometric analysis is used for this purpose. It has been proved that this scheme yields solutions that satisfy a global space-time discrete maximum principle unconditionally. A partitioned approach has also been proposed. This strategy reduces the computational cost of the scheme, while it preserves all stability properties. A regularization of this stabilization operator has also been developed. As for the first order finite element method, it improves the nonlinear convergence without harming the stability properties. An extension to Euler equations has also been pursued. In this case, instead of monotonicity-preserving, the developed scheme is local bounds preserving. Following the previous works, a regularized differentiable version has also been proposed. In addition, a continuation method using the parameters introduced for the regularization has been used. In this case, not only the nonlinear convergence is improved, but also the robustness of the method. However, the improvement in nonlinear convergence is limited to moderate tolerances and it is not as notable as for the scalar problem. Finally, the stabilized schemes proposed had been adapted to adaptive mesh refinement discretizations. In particular, nonconforming hierarchical octree-based meshes have been used. Using these settings, the efficiency of solving a monotonicity-preserving high-order stiff nonlinear problem has been assessed. Given a specific accuracy, the computational time required for solving the high-order problem is compared to the one required for solving a low-order problem (easy to converge) in a much finer adapted mesh. In addition, an error estimator based on the stabilization terms has been proposed and tested. The performance of all proposed schemes has been assessed using several numerical tests and solving various benchmark problems. The obtained results have been commented and included in the dissertation.La present tesi tracta sobre mètodes d'elements finits que preserven la monotonia per a problemes hiperbòlics. Concretament, els problemes que s'han utilitzat com a model en el desenvolupament d'aquesta tesi són l'equació escalar de convecció-difusió-reacció i les equacions d'Euler. Per a problemes escalars s'ha proposat un nou mètode d'estabilització mitjançant difusió artificial. S'ha provat que amb aquesta tècnica les solucions obtingudes són monòtones, l'esquema "disminueix els extrems locals", i preserva la linearitat. Aquestes propietats s'han pogut demostrar per múltiples dimensions i per malles generals. Per contra, aquests resultats només són vàlids per elements finits Lagrangians de primer ordre. També s'ha proposat una modificació de l'operador d'estabilització per tal de que aquest sigui diferenciable. Aquesta regularització ha permès millorar la convergència no-lineal notablement, mentre que les propietats d'estabilització no s'han vist alterades. L'anterior mètode d'estabilització s'ha adaptat a discretitzacions d'alt ordre. Concretament, s'ha utilitzat anàlisi isogeomètrica en espai i temps per a aquesta tasca. S'ha provat que les solucions obtingudes mitjançant aquest mètode satisfan el principi del màxim discret de forma global. També s'ha proposat un esquema particionat. Aquesta alternativa redueix el cost computacional, mentre preserva totes les propietats d'estabilitat. En aquest cas, també s'ha realitzat una regularització de l'operador d'estabilització per tal de que sigui diferenciable. Tal i com s'ha observat en els mètodes de primer ordre, aquesta regularització permet millorar la convergència no-lineal sense perdre les propietats d'estabilització. Posteriorment, s'ha estudiat l'adaptació dels mètodes anteriors a les equacions d'Euler. En aquest cas, en comptes de preservar la monotonia, l'esquema preserva "cotes locals". Seguint els desenvolupaments anteriors, s'ha proposat una versió diferenciable de l'estabilització. En aquest cas, també s'ha desenvolupat un mètode de continuació utilitzant els paràmetres introduïts per a la regularització. En aquest cas, no només ha millorat la convergència no-lineal sinó que l'esquema també esdevé més robust. Per contra, la millora en la convergència no-lineal només s'observa per a toleràncies moderades i no és tan notable com en el cas dels problemes escalars. Finalment, els esquemes d'estabilització proposat s'han adaptat a malles de refinament adaptatiu. Concretament, s'han utilitzat malles no-conformes basades en octrees. Utilitzant aquesta configuració, l'eficiència de resoldre un problema altament no-lineal ha estat avaluada de la següent forma. Donada una precisió determinada, el temps computacional requerit per resoldre el problema utilitzant un esquema d'alt ordre ha estat comparat amb el temps necessari per resoldre'l utilitzant un esquema de baix ordre en una malla adaptativa molt més refinada. Addicionalment, també s'ha proposat un estimador de l'error basat en l'operador d'estabilització. El comportament de tots els esquemes proposats anteriorment s'ha avaluat mitjançant varis tests numèrics. Els resultats s'han compilat i comentat en la present tesi.Postprint (published version