Adaptive domain decomposition algorithms and finite volume/finite element approximation for advection-diffusion equations

Poor performances can be obtained from classical domain decomposition algorithms to solve advection-diffusion equations in the case of convection dominated flows. Therefore, adaptive domain decomposition have been developed for such flows. We investigate the properties of some algorithms of this kind in the framework of a finite volume/finite element discretization.This research was carried out while the author was visiting the Group of Applied Mathematics and Simulation of CRS4, and was supported by an HCM fellowship.     

The Optimisation of the Mesh in First-Order Systems Least-Squares Methods

We describe an algorithm for optimising the mesh in the least-squares finite element discretisation of first-order systems of partial differential equations. The key feature of the method is that the optimisation process is based entirely on the solution of local PDE problems. We apply the algorithm to the Stokes equations for the flow of a viscous incompressible fluid, and to a convection diffusion equation where convection dominates.     

Decentrage et elements finis mixtes pour les equations de diffusion-convection

We analyze a numerical scheme for scalar diffusion-convection equations. The convective term is approximated by an upwind scheme for discontinuous finite elements and the diffusion term is approximated by a mixed finite element method. Studying large convection problems, we calculate estimates which remain valid when the diffusion term vanishes. Since the error analysis shows that the convection term is approximated less precisely than the diffusion term, the initial formulation is modified in order to balance errors from these two terms.      

Characteristic mixed finite element approximation of transient convection diffusion optimal control problems

In this paper, we investigate a characteristic mixed finite element approximation of transient convection diffusion optimal control problems. The state and the adjoint state are approximated by characteristic mixed finite element method, while the control is discretized by standard finite element method. We derive the continuous and discrete first-order optimality conditions and prove a priori error estimates for the state, the adjoint state and the control. Numerical examples are presented to illustrate the theoretical findings.     

An assessment of discretizations for convection-dominated convection–diffusion equations

The performance of several numerical schemes for discretizing convection-dominated convection–diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov–Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.     

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

We derive in this paper guaranteed and fully computable a posteriori error estimates for vertex-centered finite-volume-type discretizations of transient linear convection–diffusion–reaction equations. Our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew-symmetric part of the differential operator. Lower bounds, global-in-space but local-in-time, are also derived. These lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting. On the basis of the derived estimates, we propose an adaptive algorithm which enables to automatically achieve a user-given relative precision. This algorithm also leads to efficient calculations as it balances the time and space error contributions. As an example, we apply our estimates to the combined finite volume–finite element scheme, including such features as use of mass lumping for the time evolution or reaction terms, of upwind weighting for the convection term, and discretization on nonmatching meshes possibly containing nonconvex and non-star-shaped elements. A collection of numerical experiments illustrates the efficiency of our estimates and the use of the space–time adaptive algorithm.     

Automated interrogation and adaptive subdivision of shape using medial axis transform

We present a method for the generation of coarse and fine finite element meshes on multiply connected surfaces. Our method is based on the medial axis transform (MAT) which is employed to decompose a complex shape into topologically simple subdomains. One important property of our approach is that MAT is effectively employed to automatically extract some important shape characteristics and their length scales. Using this technique, we can create a coarse subdivision of a complex surface and select local element size to generate fine triangular meshes within individual subregions. The MAT allows us to carry out these processes in an automated manner. Thus, our approach can lead to integration of automated finite element (FE) mesh generation schemes into existing FE preprocessing systems. We also briefly discuss several design and analysis applications, which include adaptive surface approximations and adaptive h- and p-version finite element analysis (FEA) processes, in order to demonstrate our method.     

Finite element approximations of nonlinear eigenvalue problems in quantum physics

In this paper, we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics. We derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations. In particular, we analyze the convergence and complexity of an adaptive finite element method. In our analysis, we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations. We also present several numerical examples in quantum physics that support our theory.     

On the adaptive finite element method of steady-state rolling contact for hyperelasticity in finite deformations

In this paper, we present an adaptive finite element method for steady-state rolling contact in finite deformations along with a residual based a posteriori error estimator for rolling contact problem with Coulomb friction. A general formulation of rolling contact geometry is derived from the point of view of differential geometry. Solvability conditions for the rolling contact problems are discussed. We use Newton's method to solve variational equations derived from a penalty regularization of the finite element approximation of the rolling contact problem. We provide a numerical example to illustrate the method.     

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

In this paper we present a high-order Lagrangian-decoupling method for the unsteady convection diffusion and incompressible Navier-Stokes equations. The method is based upon Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem, implicit high-order backward-differentiation finite difference schemes for integration along characteristics, finite element or spectral element spatial discretizations and mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high-order accuracy and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.     

The expanded upwind-mixed method on changing meshes for positive semi-definite problem of two-phase miscible flow

The expanded upwind-mixed method on dynamically changing meshes is presented for the positive semi-definite problem of two-phase miscible flow in porous media. The pressure is approximated by a mixed finite element method and the concentration is approximated by a method which upwinds the convection and incorporates diffusion using an expanded mixed finite element method. When the mesh changes, error estimate is derived under the assumption of only a positive semi-definite diffusion coefficient. Finally, the numerical experiments are given.     

3D finite element meshing from imaging data

This paper describes an algorithm to extract adaptive and quality 3D meshes directly from volumetric imaging data. The extracted tetrahedral and hexahedral meshes are extensively used in the finite element method (FEM). A top-down octree subdivision coupled with a dual contouring method is used to rapidly extract adaptive 3D finite element meshes with correct topology from volumetric imaging data. The edge contraction and smoothing methods are used to improve mesh quality. The main contribution is extending the dual contouring method to crack-free interval volume 3D meshing with boundary feature sensitive adaptation. Compared to other tetrahedral extraction methods from imaging data, our method generates adaptive and quality 3D meshes without introducing any hanging nodes. The algorithm has been successfully applied to constructing quality meshes for finite element calculations.     

An adaptive method for the Stefan problem and its application to endoglacial conduits

This paper concerns an adaptive finite element method for the Stefan one-phase problem. We derive a parabolic variational inequality using the Duvaut transformation. In each time-step we consider an adaptive algorithm based on a combination of the Uzawa method associated with the corresponding multivalued operator and a convergent adaptive method for the linear problem. We justify the convergence of the method. As an application we model an endoglacial conduit in which a phase change phenomenon takes place.     

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators.     

Adaptive error control for multigrid finite element

We consider the problem of adaptive error control in the finite element method including the error resulting from, inexact solution of the discrete equations. We prove a posteriori error estimates for a prototype elliptic model problem discretized by the finite element with a canomical multigrid algorithm. The proofs are based on a combination of so-called strong stability and, the orthogonality inherent in both the finite element method can the multigrid algorithm.     

Adaptive finite element heterogeneous multiscale method for homogenization problems

In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.     

An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions. The model is formulated as a boundary value problem for the Helmholtz equation with a transparent boundary condition. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated Dirichlet-to-Neumann boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. A new adaptive finite element algorithm is proposed for solving the acoustic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.     

On the Stability of Continuous–Discontinuous Galerkin Methods for Advection–Diffusion–Reaction Problems

We consider a finite element method which couples the continuous Galerkin method away from internal and boundary layers with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. The stability properties of the coupled method are illustrated with a numerical experiment.     

Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation

In this paper we first review our recent work on a new framework for adaptive turbulence simulation: we model turbulence by weak solutions to the Navier–Stokes equations that are wellposed with respect to mean value output in the form of functionals, and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output. We then derive a local energy estimate for a particular finite element method, which we connect to related work on dissipative weak Euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions. The ideas are illustrated by numerical results, where we observe a law of finite dissipation with respect to a decreasing mesh size.     

Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in $H^1$-norm and for pressure in $L^2$-norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results.