Martin Vohralík

A Posteriori Error Estimates for Lowest-Order Mixed Finite Element Discretizations of Convection-Diffusion-Reaction Equations

We establish residual a posteriori error estimates for lowest-order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations on simplicial meshes in two or three space dimensions. The upwind-mixed scheme is considered as well, and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion-dispersion tensor and on a possible convection dominance. Global upper bounds for the approximation error in the energy norm are derived, where in particular all constants are evaluated explicitly, so that the estimators are fully computable. Our estimators give local lower bounds for the error as well, and they hold from the cases where convection or reaction are not present to convection- or reaction-dominated problems; we prove that their local efficiency depends only on local variations in the coefficients and on the local Peclet number. Moreover, the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies. The main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element, known in the mixed finite element method, and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem. Numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators.

Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems

We propose and study a posteriori error estimates for convection-diffusion-reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior-penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using H(div,@W)-conforming diffusive and convective flux reconstructions, thereby extending the previous work on pure diffusion problems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Peclet and Damkohler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results.

A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.

A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers

For the finite volume discretization of a second-order elliptic model problem, we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be bounded by constructing an equilibrated Raviart-Thomas-Nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector. Next, claiming that the discretization error and the algebraic one should be in balance, we construct stopping criteria for iterative algebraic solvers. An attention is paid, in particular, to the conjugate gradient method which minimizes the energy norm of the algebraic error. Using this convenient balance, we also prove the efficiency of our a posteriori estimates; i.e., we show that they also represent a lower bound, up to a generic constant, for the overall energy error. A local version of this result is also stated. This makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error. Numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers.

Adaptive Inexact Newton Methods with A Posteriori Stopping Criteria for Nonlinear Diffusion PDEs

We consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations of diffusion type. To solve these systems, some iterative nonlinear solver and, on each step of this solver, some iterative linear solver are used. We derive adaptive stopping criteria for both iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely, the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error no longer affects the overall error significantly. Our estimates also yield a guaranteed upper bound on the overall error at each step of the nonlinear and linear solvers. We prove the (local) efficiency and robustness of the estimates with respect to the size of the nonlinearity owing, in particular, to the error measure involving the dual norm of the residual. Our developments hinge on equilibrated flux reconstructions and yiel...

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.

Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients

We study in this paper a posteriori error estimates for H1-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

ABSTRACT We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H 1(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H 1(Ω) consist of functions from H 1 on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.

A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation

We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, various finite volume, and mixed finite element methods in space and the backward Euler scheme in time. Extensions to conforming and nonconforming finite element spatial discretizations are also outlined. Our estimates are based on a

A Framework for Robust A Posteriori Error Control in Unsteady Nonlinear Advection-Diffusion Problems

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