Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa

AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20


Published: SIAM Review, 44 (2002) 631--658.

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Convergence of Adaptive Finite Element Methods

This paper explores the approximation power of Moving Least-Squares (MLS) approximations in the context of higher-order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: (1) computation of high-order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, (2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and (3) multiresolution shock detection and selective limiting. A major advantage of the proposed methodology over the most popular existing higher-order methods is related to the viscous discretization. The use of MLS approximations allows the direct reconstruction of high-order viscous fluxes using quite compact stencils, and without introducing new degrees of freedom, which results in a significant reduction in storage and workload. A selective limiting procedure is proposed, based on the multiresolution properties of the MLS approximants, which allows to switch off the limiters in smooth regions of the flow. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest.

Finite volume solvers and movingleast-squares approximations for thecompressible Navier-Stokes equations onunstructured grids

A fully coupled electromagnetic-thermal-mechanical model for coalbed methane extraction with microwave heating

Numerical analysis of microwave heating cavity: Combining electromagnetic energy, heat transfer and fluid dynamics for a NaY zeolite fixed-bed

Numerical analysis and optimization of the microwave inductive heating performance of water film

Simulation and control of continuous glass melting by microwave heating in a single-mode cavity with energy efficiency optimization

A very high-order finite volume method based on weighted least squares for elliptic operators on polyhedral unstructured grids

An a posteriori error estimate suitable for finite-volume adaptive computations is presented. The error estimate combines the least-squares method regressions with the residual computation, which provides information from the grid quality and the governing equations for a better local adaptation of the unstructured grid. The decision algorithm uses the information provided by the error estimate and does not require problem-dependent constants; it also uses a grid interface correction step to provide a smoother and a high-quality adaptive grid. The proposed error estimate and the adaptive refinement algorithm are verified against analytic solution for different two-dimensional problems. In addition, calculations of three-dimensional laminar flows with different types of unstructured grids have demonstrated the applicability of the adaptive method.

Residual Least-Squares Error Estimate for Unstructuredh-Adaptive Meshes

In this study, the continuous microwave production of (Pr, Zr)SiO 2 and (Ti, Sb, Cr)O 2 pigments was modelled. The mathematical model includes a two-way coupling between the Maxwell’s equations and the heat equation as well as the coupling between the thermal and the chemical interfaces. The COMSOL Multiphysics software was used together with a specially designed MATLAB controller developed to automatically manage the input power and the cavity impedance. A single-mode cavity operating at 2.45 GHz, together with a rectangular waveguide, was used. A chemical model using experimental data was implemented using a model-fitting approach to predict the chemical conversion with time. Results show that the model-fitting method yields a good agreement with the experimental data. Attenuation occurs significantly within the material as the dielectric properties have high values. As such, prohibitive temperatures can be attained easily, although this was taken into account by the developed controller to manage the input power properly. The electromagnetic efficiency is highly dependent on the thermal field, making the boundary convection coefficient the parameter that affects the most the electromagnetic heating performance. A parametric study, that includes the variation of velocity and convection coefficient, was performed to access the impact of these variables in the efficiency of the process. The results show that increasing the velocity leads to a reduction of the maximum temperature, improving the electromagnetic efficiency. Moreover, the green-house gas specific emissions are lower when compared to the conventional process. The thermal conductivity is also a crucial parameter that can contribute to the electric field attenuation and the superficial heating.

Duarte M.S. Albuquerque

Papers

Simulation and control of continuous glass melting by microwave heating in a single-mode cavity with energy efficiency optimization

A very high-order finite volume method based on weighted least squares for elliptic operators on polyhedral unstructured grids

Residual Least-Squares Error Estimate for Unstructuredh-Adaptive Meshes

Numerical simulation and optimization of the ceramic pigments production process using microwave heating

On the use of polyhedral unstructured grids with a moving immersed boundary method