High-order Finite Element Approximation Research Articles

High-order splitting finite element methods for the subdiffusion equation with limited smoothing property

In contrast with the diffusion equation which smoothens the initial data to C ∞ C^\infty for t > 0 t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.

Uniform preconditioners for high order finite element approximations of planar linear elasticity

A new overlapping Additive Schwarz preconditioner is developed for high order finite element approximation of planar linear elastic problems on triangular meshes. The new preconditioner results in a condition number that is bounded independently of the degree p p , the mesh-size h h and the ratio λ / μ \lambda /\mu . The resulting condition number is reduced to roughly 6.0 6.0 for all values of the parameters and discretization parameters on standard test problems. Crucially, the overall cost of the new preconditioner is comparable to the cost of applying standard domain decomposition based preconditioners.

Optimal sensor placement and estimator-based temperature control for a deep drawing process

Deep drawing is one of the most important forming processes for the forming of flat sheet blanks, where the formation of wrinkles and the appearance of cracks can be a problem, especially in areas of high geometric complexity. The local increase of the temperature in these critical areas can help to improve the formability of the material and thus reduce defects. The present paper aims at a targeted temperature control of the die of a deep-drawing mold. For this sensors are placed systematically to develop an estimator for the spatial–temporal temperature evolution to subsequently realize tracking control using the embedded actuation devices. A continuum representation of the temperature distribution in the die is derived and transferred to a high order finite element (FE) approximation to take the complex-shaped geometry of the tool into account. Parameter identification is performed based on measurement data to improve the accuracy of the FE approximation and model order reduction (MOR) techniques are applied to determine a sufficiently low order system representation. A mixed-integer optimization problem is formulated and solved making use of different formulations of the observability Gramian to determine the optimal sensor locations and a Kalman filter is designed as an estimator based on a reduced order model. Moreover, a linear-quadratic regulator with integral part combined with the Kalman filter is developed to react efficiently towards disturbances. Finally this theoretical framework is tested in a real experiment.

Selected aspects of constrained conforming approximation in higher-order FEM

This paper responds to numerous questions and inquiries related to constrained approximation (approximation with hanging nodes) in higher-order finite element methods that we received during the ESCO 2022 conference. We discuss explicit constraints enforced on linear algebra level as well as implicit constraints which are built into the basis functions of the finite element space, and selected additional mathematical and algorithmic aspects of conforming higher-order finite element approximations with one-level and arbitrary-level hanging nodes.

Preconditioning the Mass Matrix for High Order Finite Element Approximation on Tetrahedra

A preconditioner for the mass matrix for high order finite element discretization on tetrahedra is presented and shown to give a condition number that is independent of both the mesh size and the polynomial order of the elements. The preconditioner is described in terms of a new, high order basis which has the usual property whereby individual functions are associated with distinct geometric entities of the tetrahedron. It is shown that the basis enjoys the novel property that the resulting mass matrix is spectrally equivalent to its own diagonal with constants independent of h and p. Although the exposition is based on an explicit basis, the preconditioner can be applied to any choice of basis. In particular, the basis can be used to specify a basis-independent additive Schwarz method, meaning that, in order to apply the preconditioner to an alternative basis, one only need to implement an appropriate change of basis.

Optimal Sensor Placement for Temperature Control in a Deep Drawing Tool

Deep drawing is one of the most important forming processes for shaping flat sheet blanks. One problem of this technique is the formation of wrinkles and the occurrence of cracks especially in areas of high geometric complexity. The local increase of the temperature in critical areas can help to increase the flowability and thus reduce defects. This requires to include sensors and actuators at appropriate locations in tool. The present paper aims at systematically placing sensors in the die of a specifically manufactured deep-drawing mold to develop an observer for the spatial-temporal temperature evolution. The latter forms the basis for targeted intervention by built-in actuators. A continuum representation of the temperature distribution in the die is derived and transferred to a high order finite element (FE) approximation to take the complex-shaped geometry of the tool into account. The model is segmented and model order reduction (MOR) techniques are applied to determine a sufficiently low order system representation that is applicable for optimal sensor placement. For this, a mixed-integer optimization problem is formulated and solved making use of different reduced-order formulations of the observability Gramian and suitable measures for interpretation. Furthermore a Kalman filter is implemented based on a reduced order model to evaluate the performance of the different sensor placements.

Mesh quality metrics for isogeometric Bernstein–Bézier discretizations

High-order finite element methods harbor the potential to deliver improved accuracy per degree of freedom versus low-order methods. Their success, however, hinges upon the use of a curvilinear mesh of not only sufficiently high accuracy but also sufficiently high quality. In this paper, theoretical results are presented quantifying the impact of mesh parameterization on the accuracy of a high-order finite element approximation, and a formal definition of shape regularity is introduced for curvilinear meshes based on these results. This formal definition of shape regularity in turn inspires a new set of quality metrics for curvilinear finite elements. Computable bounds are established for these quality metrics using the Bernstein–Bézier form, and a new curvilinear mesh optimization procedure is proposed based on these bounds. Numerical results confirming the importance of shape regularity in the context of high-order finite element methods are presented, and numerical results demonstrating the promise of the proposed curvilinear mesh optimization procedure are also provided. The theoretical results in this paper apply to any piecewise-polynomial or piecewise-rational finite element method posed on a mesh of polynomial or rational mapped simplices and hypercubes. As such, they apply not only to classical continuous Galerkin finite element methods but also to discontinuous Galerkin finite element methods and even isogeometric methods based on NURBS, T-splines, or hierarchical B-splines.

Higher-order finite element approximation of the dynamic Laplacian

The dynamic Laplace operator arises from extending problems of isoperimetry from fixed manifolds to manifolds evolved by general nonlinear dynamics. Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physically manifest in fluid flows as jets, vortices, and more complicated structures. Two robust and efficient finite-element discretisation schemes for numerically computing the dynamic Laplacian were proposed in Froyland and Junge [SIAM J. Appl. Dyn. Syst. 17 (2018) 1891–1924]. In this work we consider higher-order versions of these two numerical schemes and analyse them experimentally. We also prove the numerically computed eigenvalues and eigenvectors converge to the true objects for both schemes under certain assumptions. We provide an efficient implementation of the higher-order element schemes in an accompanying Julia package.

Hierarchical high order finite element spaces in [formula omitted] for a stabilized mixed formulation of Darcy problem

The classical dual mixed finite element method for flow simulations is based on H(div,Ω) conforming approximation spaces for the flux, which guarantees continuous normal components on element interfaces, and discontinuous approximations in L2(Ω) for the pressure. However, stability and convergence can only be obtained for compatible approximation spaces. Stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance. The main purpose of this paper is to present a high-order finite element methodology to solve the Darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of H(div,Ω) and H1(Ω). The chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables, without requiring any compatibility constraint, as required in mixed methods for these problems. Convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with Raviart–Thomas elements.

Implementation of Numerical Integration to High-Order Elements on the GPUs

This article presents ways to implement a resource-consuming algorithm on hardware with a limited amount of memory, which is the GPU. Numerical integration for higher-order finite element approximation was chosen as an example algorithm. To perform computational tests, we use a non-linear geometric element and solve the convection-diffusion-reaction problem. For calculations, a Tesla K20m graphics card based on Kepler architecture and Radeon r9 280X based on Tahiti XT architecture were used. The results of computational experiments were compared with the theoretical performance of both GPUs, which allowed an assessment of actual performance. Our research gives suggestions for choosing the optimal design of algorithms as well as the right hardware for such a resource-demanding task.

Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting

This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems.

On using geometry-based high-order finite element approximations for mid-frequency response of fluid-loaded elastic structures

Accurate solution of the exterior scattering and radiation problems in the mid-frequency regime remains of high interest and a computationally challenging task. For wave-dependent problems with large dispersion errors, high-order approximations provide several advantages, including increased accuracy and error convergence rate compared to low-order methods. The effective use of high-order finite element type approximations require special attention to geometry. In particular, one must use high-order (curved) meshes in order to preserve the accuracy and convergence advantages. We present recent developments in the use of geometry-based high-order approximations for computing scattering response of fluid-loaded elastic structures submerged in an acoustic medium. We present examples with validation against experimental data to demonstrate the effectiveness of these techniques as well as highlight future technical challenges to push to the higher ends of the mid-frequency regime.Accurate solution of the exterior scattering and radiation problems in the mid-frequency regime remains of high interest and a computationally challenging task. For wave-dependent problems with large dispersion errors, high-order approximations provide several advantages, including increased accuracy and error convergence rate compared to low-order methods. The effective use of high-order finite element type approximations require special attention to geometry. In particular, one must use high-order (curved) meshes in order to preserve the accuracy and convergence advantages. We present recent developments in the use of geometry-based high-order approximations for computing scattering response of fluid-loaded elastic structures submerged in an acoustic medium. We present examples with validation against experimental data to demonstrate the effectiveness of these techniques as well as highlight future technical challenges to push to the higher ends of the mid-frequency regime.

A modified graded mesh and higher order finite element approximation for singular perturbation problems

A singularly perturbed convection diffusion problem is solved numerically using finite element method based on higher order polynomials. More precisely, we introduce a modified graded mesh generated using some implicitly defined functions. Higher order parameter uniform convergence is obtained in ϵ-weighted energy norm. Moreover, the error estimates obtained are optimal in the sense that they are free from logarithmic factor. A number of test examples are taken into account and a rigorous comparative analysis is presented. Moreover, we compare the proposed method with others found in the literature.

New Analysis of Galerkin FEMs for Miscible Displacement in Porous Media

The paper is concerned with optimal error estimates of classical Galerkin FEMs for the equations of incompressible miscible flows in porous media. The analysis done in the last several decades shows that classical Galerkin FEMs provide the numerical concentration of the accuracy $$O(\tau ^k+h^{r+1} + h^s)$$ in $$L^2$$ -norm. This analysis leads to the use of higher order finite element approximation to the pressure than that to the concentration in various numerical simulations to achieve the best rate of convergence. However, this error estimate is not optimal. The purpose of this paper is to establish the optimal $$L^2$$ error estimate $$O(\tau ^k + h^{r+1} + h^{s+1})$$ , from which one can see that the best convergence rate can be achieved by taking the same order $$(r=s$$ ) approximation to the concentration and pressure. Clearly Galerkin FEMs with $$r=s$$ are less expensive in computation and easier for implementation. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis.

Preconditioning the Mass Matrix for High Order Finite Element Approximation on Triangles

The problem of preconditioning the $p$-version mass matrix on meshes of (possibly curvilinear) triangular elements in two dimensions is considered. Through a judicious choice of hierarchical basis,...

An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell’s equations

In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell’s equations in waveguide configurations. The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell’s equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations.

Shape optimization using the cut finite element method

We present a cut finite element method for shape optimization in the case of linear elasticity. The elastic domain is defined by a level-set function, and the evolution of the domain is obtained by moving the level-set along a velocity field using a transport equation. The velocity field is the largest decreasing direction of the shape derivative that satisfies a certain regularity requirement and the computation of the shape derivative is based on a volume formulation. Using the cut finite element method no re-meshing is required when updating the domain and we may also use higher order finite element approximations. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements at the boundary, which provides control of the variation of the solution in the vicinity of the boundary. We implement and illustrate the performance of the method in the two-dimensional case, considering both triangular and quadrilateral meshes as well as finite element spaces of different order.

Estimating and localizing the algebraic and total numerical errors using flux reconstructions

This paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization and an arbitrary iterative algebraic solver. The derived bounds are based on the flux reconstruction techniques, do not contain any unspecified constants, and allow estimating the local distribution of both errors over the computational domain. We also discuss bounds on the discretization error, their application for constructing mathematically justified stopping criteria for iterative algebraic solvers, global and local efficiency of the total error upper bound, and the relationship to the previously published estimates on the algebraic error. Theoretical results are illustrated on numerical experiments for higher-order finite element approximations and the preconditioned conjugate gradient method. They in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over thecomputational domain and illustrate the associate cost.

Errata to “Approximation classes for adaptive higher order finite element approximation”

Errata to “Approximation classes for adaptive higher order finite element approximation”

Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications

We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.