Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic p-Laplacian

We generalize the a posteriori techniques for the linear heat equation in [Ver03] to the case of the nonlinear parabolic p-Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of the error and the residual, which is the key property for proving the error bounds

A Posteriori Error Analysis of the Method of Characteristics

We consider a two- or three-dimensional timedependent diusion-convection-reaction problem and its discretization by the method of characteristics and standard nite elements. We perform the a posteriori error analysis of this discretization and prove optimal error estimates, which lead to an ecient adaptivity strategy both for the time step and the spatial mesh. The estimates are robust with respect to the ratios of the diusion to the reaction or convection. Some numerical experiments support the theoretical results

An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem

AbstractAn a posteriori upper bound is derived for the nonstationary convection–diffusion problem using the Crank–Nicolson scheme and continuous, piecewise linear stabilized finite elements with large aspect ratio. Following Lozinski et al. (2009) [13], a quadratic time reconstruction is used.A space and time adaptive algorithm is developed to ensure the control of the relative error in the L2(H1) norm. Numerical experiments illustrating the efficiency of this approach are reported; it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step, even in the convection dominated regime and in the presence of boundary layers

Simulación de la calidad de aire con dispersión muy anisotrópica mediante elementos finitos adaptativos

En aquest document es simula el problema de qualitat d'aire al voltant d'un emissor puntual en condicions atmosf eriques de calma mitjan cant Elements Finits adaptatius. En concret s'aplica al cas de La Oroya (Per u). La resoluci o del problema d'advecció-difusi o-reacci ó mitjan cant Elements Finits acostuma a produir oscilacions en la soluci o. Per tal d'aminorar-les i poder aconseguir les corbes d'inmissi o (que t picament prenen valors varis ordres de magnitud inferiors a la concentraci o emesa) no n'hi ha prou amb la utilitzaci o d'esquemes estabilitzats. Per aix o es proposa emprar un proc es adaptatiu mitjan cant el qual es canvia la discretitzaci o espacial en el rang d'inter es de la solucio. S'estudia el comportament de la soluci o i es proposa un indicador de l'error especialment dissenyat pel problema d'emissors puntuals que delimita les zones on es produeixen les oscilacions. S'utilitza un esquema de remallat basat en la imposici o d'un volum m axim als elements de certes regions. Aquest algoritme permet evitar recalcular cada interval de temps ja que es possible augmentar en una sola iteració o la densitat d'elements en una region

An adaptive SUPG method for evolutionary convection-diffusion equations

An adaptive algorithm for the numerical simulation of time-dependent convection-diffusion-reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convection-diffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from [18] will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in [6]

A Residual Based A Posteriori Error Estimators for AFC Schemes for Convection-Diffusion Equations

In this work, we propose a residual-based a posteriori error estimator for algebraic flux-corrected (AFC) schemes for stationary convection-diffusion equations. A global upper bound is derived for the error in the energy norm for a general choice of the limiter, which defines the nonlinear stabilization term. In the diffusion-dominated regime, the estimator has the same convergence properties as the true error. A second approach is discussed, where the upper bound is derived in a posteriori way using the Streamline Upwind Petrov Galerkin (SUPG) estimator proposed in \cite{JN13}. Numerical examples study the effectivity index and the adaptive grid refinement for two limiters in two dimensions

Study On Covolume-Upwind Finite Volume Approximations For Linear Parabolic Partial Differential Equations

In this thesis we solve two-dimensional linear parabolic partial differential equations with pure Dirichelet boundary conditions, using the bilinear covolume-upwind finite volume method on rectangular grids to discretize the spatial variables and the Crank-Nicholson method for the time variable. These PDEs provide a model for problems from various fields of engineering and applied sciences, such as unsteady viscous flow problems, the simulation of oil extraction from underground reservoirs, transport of air and ground water pollutants and modeling of semiconductor devices. Finite volume method has the important advantage of allowing the conversion of integrations over the control volume to integrations over its boundary based on Green\u27s Theorem. Then, one can use quadrature rules to approximate the resulting integrals. In order to avoid non-physical oscillations that can arise from the numerical solution of convection-dominated problems when using the central finite volume scheme, we generate non-standard control volumes using local Peclet\u27s numbers and the upwind principle. We numerically compare the covolume-upwind finite volume method with the central and the upwind finite volume schemes, demonstrating stability and better convergence of the method through various examples