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

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the alpha-th order (1 <= \alpha <= k+1) divided difference of the DG error in the L2-norm is of order k+(3-alpha)/2 when upwind fluxes are used, under the condition that |f'(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order k+(3-alpha)/2 in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least (3k/2+1)th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order k+1 in the L2-norm for the divided differences of DG errors and thus (2k+1)th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results

Adaptive hybrid discontinuous methods for fluid and wave problems

This PhD thesis proposes a p-adaptive technique for the Hybridizable Discontinuous Galerkin method (HDG). The HDG method is a novel discontinuous Galerkin method (DG) with interesting characteristics. While retaining all the advantages of the common DG methods, such as the inherent stabilization and the local conservation properties, HDG allows to reduce the coupled degrees of freedom of the problem to those of an approximation of the solution de¿ned only on the faces of the mesh. Moreover, the convergence properties of the HDG solution allow to perform an element-by-element postprocess resulting in a superconvergent solution. Due to the discontinuous character of the approximation in HDG, p-variable computations are easily implemented. In this work the superconvergent postprocess is used to de¿ne a reliable and computationally cheap error estimator, that is used to drive an automatic adaptive process. The polynomial degree in each element is automatically adjusted aiming at obtaining a uniform error distribution below a user de¿ned tolerance. Since no topological modi¿cation of the discretization is involved, fast adaptations of the mesh are obtained. First, the p-adaptive HDG is applied to the solution of wave problems. In particular, the Mild Slope equation is used to model the problem of sea wave propagation is coastal areas and harbors. The HDG method is compared with the continuous Galerkin (CG) ¿nite element method, which is nowadays the common method used in the engineering practice for this kind of applications. Numerical experiments reveal that the e¿ciency of HDG is close to CG for uniform degree computations, clearly outperforming other DG methods such as the Compact Discontinuous Galerkin method. When p-adaptivity is considered, an important saving in computational cost is shown. Then, the methodology is applied to the solution of the incompressible Navier-Stokes equations for the simulation of laminar ¿ows. Both steady state and transient applications are considered. Various numerical experiments are presented, in 2D and 3D, including academic examples and more challenging applications of engineering interest. Despite the simplicity and low cost of the error estimator, high e¿ciency is exhibited for analytical examples. Moreover, even though the adaptive technique is based on an error estimate for just the velocity ¿eld, high accuracy is attained for all variables, with sharp resolution of the key features of the ¿ow and accurate evaluation of the ¿uid-dynamic forces. In particular, high degrees are automatically located along boundary layers, reducing the need for highly distorted elements in the computational mesh. Numerical tests show an important reduction in computational cost, compared to uniform degree computations, for both steady and unsteady computations

A review of recent advances in discretization methods, a posteriori error analysis, and adaptive algorithms for numerical modeling in geosciences

International audienceTwo research subjects in geosciences which lately underwent significant progress are treated in this review. In the first part we focus on one key ingredient for the numerical approximation of the Darcy flow problem, namely the discretization of diffusion terms on general polygonal/polyhedral meshes. We present different schemes and discuss in detail their fundamental numerical properties such as stability, consistency, and robustness. The second part of the paper is devoted to error control and adaptivity for model geosciences problems. We present the available a posteriori estimates guaranteeing the maximal overall error and show how the different error components can be identified. These estimates are used to formulate adaptive stopping criteria for linear and nonlinear solvers, time step choice adjustment, and adaptive mesh refinement. Numerical experiments illustrate such entirely adaptive algorithms

Hierarchical finite element methods for compressible flow problems

The thesis is concerned with the introduction of the CG1-DG2 method and the design of an hp-adaptive algorithm in the context of convection-dominated problems in 2D. The CG1-DG2 method combines the continuous Galerkin (CG) method with the discontinuous Galerkin (DG) method by enriching the continuous linear finite element (CG1) space with discontinuous quadratic basis functions. The resulting finite element approximation is continuous at the vertices of the mesh but may be discontinuous across edges. Analysis of the CG1-DG2 discretization in the context of a scalar advection equation shows that the use of upwind-biased convective fluxes leads to an approximation which is stable and exhibits the same convergence rates as the quadratic discontinuous (DG2) method. However, the CG1-DG2 space has fewer degrees of freedom than the DG2 space. In the case of Poisson's equation different strategies known from the DG method can be adopted to approximate the numerical fluxes: the symmetric and non-symmetric interior penalty method as well as the Baumann-Oden method. A priori error estimates for the DG2 method can be shown to hold for the CG1-DG2 approximation as well. Numerical studies confirm that the proposed method is stable and converges at the same rate as the fully discontinuous piecewise-quadratic version. We also present an extension of the CG1-DG2 method to solve the Euler equations and show numerical results which indicate that the CG1-DG2 method gives results similar to those obtained by the DG method. The second part of this thesis presents an hp-adaptive framework for convection-dominated problems. The idea of this algorithm is to divide the mesh in smooth and non-smooth parts, where the smoothness refers to the regularity of the approximated solution. In smooth parts the polynomial degree is increased (p-adaptivity) whereas in non-smooth parts h-adaptivity for linear elements is used. Hereby, a parameter-free regularity estimator is used to determine the smoothness of a function and its gradient by comparing those with reconstructed approximations. In smooth elements the CG1-DG2 method is used. In non- smooth elements a flux-corrected transport scheme is applied and combined with h-adaptivity based on the so-called reference solution approach. Numerical experiments are performed for advection and advection-diffusion equations. Those show the advantage of the hp-adaptive algorithm over pure h-refinement in the context of FCT schemes