Anisotropic Finite Element Meshes Research Articles

On the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction–diffusion systems

Nonlinear reaction–diffusion systems are widely employed to study the spatio-temporal chaotic behavior that occurs in excitable media such as cardiac tissue where sufficiently strong perturbations can excite nonlinear propagating waves which can form spiral waves in two dimensions or scroll waves in three dimensions. The numerical simulation of these waves calls for grids that are extremely fine over the whole computational domain to accurately predict the trajectories and multiplication of wave fronts and therefore leads to huge computational challenges. Mesh adaptation methods can reduce the number of degrees of freedom required for a given accuracy but they also have a cost and it is not clear if they are competitive with respect to very fine uniform meshes. Previous mesh adaptation techniques applied to spatio-temporal chaotic behavior have been mostly limited to the two-dimensional case. The purpose of this paper is to explore the efficiency of a three-dimensional anisotropic finite element mesh adaptivity for simulating scroll wave turbulence. The computational efficiency of the proposed method is assessed using reference solutions obtained on a uniform refined mesh with more than 44 millions degrees of freedom. The proposed method reduces significantly the number of elements leading to huge saving in memory as well as in computational time. Examples of the dynamics of ventricular fibrillation in cardiac tissue will be presented illustrating the performance of the overall methodology.

The Effect of Anisotropic Mesh Adaptation on PDE-Constrained Optimal Control Problems

We address a linear-quadratic optimal control problem constrained by a scalar advection-diffusion-reaction equation. We develop an anisotropic goal-oriented error analysis to show in an optimal control framework how the employment of an anisotropic finite element mesh can reduce the computational cost. The novel ingredient is the enrichment of the a posteriori analysis, already well established in an isotropic context, with information about possible directionalities of the problem at hand. To deal with the advection dominated regime, a strongly consistent anisotropic edge-oriented symmetric stabilization is employed. This has the advantage that the optimize-then-discretize and discretize-then-optimize approaches coincide. The influence of anisotropic mesh adaptivity on the optimal control problem is extensively investigated numerically on a model test case as well as on an application related to the environmental pollution issue. The analysis is then extended to the Oseen system for a model Dirichlet control problem in fluid mechanics.

Analysis of Surface Plasmon Polariton Using Anisotropic Finite Elements

The anisotropic finite-element mesh is deployed in a vector finite-element method to numerically derive surface plasmon polaritons (SPPs) more efficiently. The superiority of the anisotropic mesh lies in the fact that SPP modes, predictably, have a sharp field variation in the surface-normal direction, whereas the variation along the surface-tangent direction is comparably gentle. A case study on metallic fiber shows that the employment of anisotropic finite elements reduces the problem size by a factor of seven.

An equilibrated residual method with a computable error approximation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in a discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both, the perturbation parameters of the problem and the anisotropy of the mesh. The equilibrated residual method has been shown to provide one of the most reliable error estimates for the reaction-diffusion problem. Its modification suggested by Ainsworth and Babuška has been proved to be robust for the case of singular perturbation. In the present work we investigate the modified method on anisotropic meshes. The method in the form of Ainsworth and Babuška is shown here to fail on anisotropic meshes. We suggest a new modification based on the stretching ratios of the mesh elements. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. Among others, the equilibrated residual method involves the solution of an infinite dimensional local problem on each element. In practical computations an approximate solution to this local problem was successfully computed. Nevertheless, up to now no rigorous analysis has been done showing the appropriateness of any computable approximation. This demands special attention since an improper approximate solution to the local problem can be fatal for the robustness of the whole method. In the present work we provide one of the desired approximations. We prove that the method is not affected by the approximate solution of the local problem.

A posteriori H1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H 1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Three modifications of these estimators are introduced and discussed. Much attention is given to the performance of the error estimator in numerical experiments. This helps to identify those estimators that are suitable for practical applications.

Approximation of the incompressible Navier–Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes

This paper describes a finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales and a pressure segregation. The former consists of adding a least-square form of the component orthogonal to the finite element space of the convective and pressure gradient terms; this allows to deal with convection-dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Likewise, we put special emphasis on the use of anisotropic grids. In particular, we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples. The preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described.

A nonconforming finite element method�with anisotropic mesh grading�for the incompressible Navier?Stokes equations�in domains with edges

Any solution of the incompressible Navier–Stokes equations in three-dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix–Raviart (nonconforming 𝒫1) elements and the pressure by piecewise constants. This method is stable for general meshes since the inf-sup condition is satisfied without minimal or maximal angle condition. The existence of solutions to the discrete problems follows. Consistency error estimates for the divergence equation are obtained for anisotropic tensor product meshes. As applications, the consistency error estimate for the Navier–Stokes solution and some discrete Sobolev inequalities are derived on such meshes. These last results provide optimal error estimates in the uniqueness case by the use of appropriately refined anisotropic tensor product meshes, namely, if Ne is the number of elements of the mesh, we prove that the optimal order of convergence h∼ Ne− 1/3.

Toward anisotropic mesh construction and error estimation in the finite element method

AbstractDirectional, anisotropic features like layers in the solution of partial differential equations can be resolved favorably by using anisotropic finite element meshes. An adaptive algorithm for such meshes includes the ingredients Error estimation and Information extraction/Mesh refinement. Related articles on a posteriori error estimation on anisotropic meshes revealed that reliable error estimation requires an anisotropic mesh that is aligned with the anisotropic solution. To obtain anisotropic meshes the so‐called Hessian strategy is used, which provides information such as the stretching direction and stretching ratio of the anisotropic elements. This article combines the analysis of anisotropic information extraction/mesh refinement and error estimation (for several estimators). It shows that the Hessian strategy leads to well‐aligned anisotropic meshes and, consequently, reliable error estimation. The underlying heuristic assumptions are given in a stringent yet general form. Numerical examples strengthen the exposition. Hence the analysis provides further insight into a particular aspect of anisotropic error estimation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 625–648, 2002; DOI 10.1002/num.10023

Application of optimally distorted finite elements for field calculation problems of electromagnetism

We applied adaptively generated anisotropic finite-element meshes (i.e., meshes containing intentionally distorted elements) for electromagnetic field calculation. By this method, we achieved significant improvement in computing performance. Moreover, anisotropic meshing proved to be an essential tool in space-time meshing. In this paper, we summarize our experiences.

Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes

Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three‐dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix–Raviart (nonconforming P1 ) elements and the pressure by piecewise constants. This method is stable for general meshes (without minimal or maximal angle condition). Denoting by Ne the number of elements in the mesh, the interpolation and consistency errors are of the optimal order h ∼ Ne−1/3 which is proved for tensor product meshes. As a by‐product, we analyse also nonconforming prismatic elements with P1 [oplus ] span {x32} as the local space for the velocity where x3 is the direction of the edge.

Some mixed finite element methods on anisotropic meshes

The paper deals with some mixed finite element methods on a class of anisotropic meshes based on tetrahedra and prismatic (pentahedral) elements. Anisotropic local interpolation error estimates are derived in some anisotropic weighted Sobolev spaces. As particular applications, the numerical approximation by mixed methods of the Laplace equation in domains with edges is investigated where anisotropic finite element meshes are appropriate. Optimal error estimates are obtained using some anisotropic regularity results of the solutions.

Crouzeix-Raviart type finite elements on anisotropic meshes

The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges. A numerical test is described.

Robust a Posteriori Error Estimation for a Singularly Perturbed Reaction–Diffusion Equation on Anisotropic Tetrahedral Meshes

We consider a singularly perturbed reaction–diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.

Edge Elements on Anisotropic Meshes and Approximation of the Maxwell Equations

This paper deals with some edge element methods on a class of anisotropic meshes based on tetrahedra and prismatic (pentahedral) elements. Anisotropic local interpolation error estimates are derived for all these types of elements and for functions from classical and weighted Sobolev spaces. As a particular application, the numerical approximation of the Maxwell equations in domains with edges is investigated using Nédélec's edge elements, where anisotropic finite element meshes are appropriate. Some anisotropic regularity results of the solutions of Maxwell equations on such domains are proved. Some numerical tests are described and inverse estimates are established, both showing that our theoretical orders of convergence are optimal.