839 research outputs found
Sharp error estimates for a discretisation of the 1D convection/diffusion equation with Dirac initial data
This paper derives sharp l and l1 estimates of the error arising from an explicit approximation of the constant coefficient 1D convection/diffusion equation with Dirac initial data. The analysis embeds the discrete equations within a semi-discrete system of equations which can be solved by Fourier analysis. The error estimates are then obtained through asymptotic approximation of the integrals resulting from the inverse Fourier transform. this research is motivated by the desire to prove convergence of approximations to adjoint partial differential equations
Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?
The contents of this paper is twofold. First, important recent results concerning finite element
methods for convection-dominated problems and incompressible flow problems are described that
illustrate the activities in these topics. Second, a number of, in our opinion, important problems in
these fields are discussed
Sharp error estimates for discretisations of the 1D convection/diffusion equation with Dirac initial data
This paper derives sharp estimates of the error arising from explicit and implicit approximations of the constant coefficient 1D convection/diffusion equation with Dirac initial data. The error analysis is based on Fourier analysis and asymptotic approximation of the integrals resulting from the inverse Fourier transform. This research is motivated by applications in computational finance and the desire to prove convergence of approximations to adjoint partial differential equations
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
A posteriori error control for discontinuous Galerkin methods for parabolic problems
We derive energy-norm a posteriori error bounds for an Euler time-stepping
method combined with various spatial discontinuous Galerkin schemes for linear
parabolic problems. For accessibility, we address first the spatially
semidiscrete case, and then move to the fully discrete scheme by introducing
the implicit Euler time-stepping. All results are presented in an abstract
setting and then illustrated with particular applications. This enables the
error bounds to hold for a variety of discontinuous Galerkin methods, provided
that energy-norm a posteriori error bounds for the corresponding elliptic
problem are available. To illustrate the method, we apply it to the interior
penalty discontinuous Galerkin method, which requires the derivation of novel a
posteriori error bounds. For the analysis of the time-dependent problems we use
the elliptic reconstruction technique and we deal with the nonconforming part
of the error by deriving appropriate computable a posteriori bounds for it.Comment: 6 figure
An asymptotic induced numerical method for the convection-diffusion-reaction equation
A parallel algorithm for the efficient solution of a time dependent reaction convection diffusion equation with small parameter on the diffusion term is presented. The method is based on a domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. Parallelism is evident at two levels. Domain decomposition provides parallelism at the highest level, and within each domain there is ample opportunity to exploit parallelism. Run time results demonstrate the viability of the method
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
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