Finite Element Error Research Articles

Optimized Schwarz methods for the coupling of cylindrical geometries along the axial direction

In this work, we focus on the Optimized Schwarz Method for circular flat interfaces and geometric heterogeneous coupling arising when cylindrical geometries are coupled along the axial direction. In the first case, we provide a convergence analysis for the diffusion-reaction problem and jumping coefficients and we apply the general optimization procedure developed in Gigante and Vergara (Numer. Math. 131 (2015) 369–404). In the numerical simulations, we discuss how to choose the range of frequencies in the optimization and the influence of the Finite Element and projection errors on the convergence. In the second case, we consider the coupling between a three-dimensional and a one-dimensional diffusion-reaction problem and we develop a new optimization procedure. The numerical results highlight the suitability of the theoretical findings.

Saturation and Reliable Hierarchical a Posteriori Morley Finite Element Error Control

This paper proves the saturation assumption for the nonconforming Morley finite element discretization of the biharmonic equation. This asserts that the error of the Morley approximation under uniform refinement is strictly reduced by a contraction factor smaller than one up to explicit higher-order data approximation terms. The refinement has at least to bisect any edge such as red refinement or 3-bisections on any triangle. This justifies a hierarchical error estimator for the Morley finite element method, which simply compares the discrete solutions of one mesh and its red-refinement. The related adaptive mesh-refining strategy performs optimally in numerical experiments. A remark for Crouzeix-Raviart nonconforming finite element error control is included.

A conforming enriched finite element method for Stokes interface problems

A new conforming enriched finite element method is presented for Stokes interface problems with interface-unfitted meshes. The conforming enriched finite element pair is constructed based on the MINI element pair. A ghost penalty term is used in the standard discretization form as a stabilization term. An inf–sup stability result is derived, which is uniform with respect to the mesh size. Our method does not limit the viscosity coefficient to be a piecewise constant. Finite element errors are proved to be optimal. Numerical experiments are carried out to validate theoretical results.

Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients

This work is concerned with the optimal control problems governed by a 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L_{{{w}^{*}}}^{2}\left( I, \mathcal M\left( {\mathit \Omega } \right) \right)$ or vector measures $\mathcal M({\mathit \Omega }, L^2(I))$. The cost functional involves the standard quadratic tracking terms and the regularization term $α\|u\|_{\mathcal M_T}$ with $α>0$. We construct and study three-level in time bilinear finite element discretizations for this class of problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.

A conforming enriched finite element method for elliptic interface problems

A new conforming enriched finite element method is presented for elliptic interface problems with interface-unfitted meshes. The conforming enriched finite element space is constructed based on the P1-conforming finite element space. Approximation capability of the conforming enriched finite element space is analyzed. The standard conforming Galerkin method is considered without any penalty stabilization term. Our method does not limit the diffusion coefficient of the elliptic interface problem to a piecewise constant. Finite element errors in H1-norm and L2-norm are proved to be optimal. Numerical experiments are carried out to validate theoretical results.

An efficient duality-based approach for PDE-constrained sparse optimization

In this paper, elliptic optimal control problems involving the $L^1$-control cost ($L^1$-EOCP) is considered. To numerically discretize $L^1$-EOCP, the standard piecewise linear finite element is employed. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discrete $L^1$-norm does not have a decoupled form. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the $L^1$-norm. It is clear that this technique will incur an additional error. To avoid the additional error, solving $L^1$-EOCP via its dual, which can be reformulated as a multi-block unconstrained convex composite minimization problem, is considered. Motivated by the success of the accelerated block coordinate descent (ABCD) method for solving large scale convex minimization problems in finite dimensional space, we consider extending this method to $L^1$-EOCP. Hence, an efficient inexact ABCD method is introduced for solving $L^1$-EOCP. The design of this method combines an inexact 2-block majorized ABCD and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block. The proposed algorithm (called sGS-imABCD) is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, but also show that our proposed algorithm is more efficient than (a) the ihADMM (inexact heterogeneous alternating direction method of multipliers), (b) the APG (accelerated proximal gradient) method.

Higher order two-scale finite element error analysis for thermoelastic problem in quasi-periodic perforated structure

In this paper, one new coupled higher order two-scale finite element method (TSFEM) for thermoelastic problem in composites is proposed. Firstly, some new two-scale asymptotic expressions and homogenization formulations for the problem are briefly given. Next, some high–low coupled approximate errors corresponding to TSFEM are analyzed. Finally, some numerical results of the displacement and the increment of temperature are presented, which show that TSFEM is an effective method for predicting the mechanical and the thermal behavior of composites in quasi-periodic perforated structure.

The Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites

The thermoelastic composites are widely used in some kinds of complex environments. In this paper, the two-scale finite element method for one class of thermal mechanical problem in periodic structure is analyzed. The two-scale finite element method and the approximate errors for one class of coupling thermoelastic problem are obtained.

Finite element contact modeling method and error characteristics of partitioned parallel computations for bullet penetration simulations

Finite element contact modeling method and error characteristics of partitioned parallel computations for bullet penetration simulations

Algorithms and analyses for stochastic optimization for turbofan noise reduction using parallel reduced-order modeling

Simulation-based optimization of acoustic liner design in a turbofan engine nacelle for noise reduction purposes can dramatically reduce the cost and time needed for experimental designs. Because uncertainties are inevitable in the design process, a stochastic optimization algorithm is posed based on the conditional value-at-risk measure so that an ideal acoustic liner impedance is determined that is robust in the presence of uncertainties. A parallel reduced-order modeling framework is developed that dramatically improves the computational efficiency of the stochastic optimization solver for a realistic nacelle geometry. The reduced stochastic optimization solver takes less than 500 s to execute. In addition, well-posedness and finite element error analyses of the state system and optimization problem are provided.

Multilevel Monte Carlo Analysis for Optimal Control of Elliptic PDEs with Random Coefficients

This work is motivated by the need to study the impact of data uncertainties and material imperfections on the solution to optimal control problems constrained by partial differential equations. We consider a pathwise optimal control problem constrained by a diffusion equation with random coefficient together with box constraints for the control. For each realization of the diffusion coefficient we solve an optimal control problem using the variational discretization [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45--61]. Our framework allows for lognormal coefficients whose realizations are not uniformly bounded away from zero and infinity. We establish finite element error bounds for the pathwise optimal controls. This analysis is nontrivial due to the limited spatial regularity and the lack of uniform ellipticity and boundedness of the diffusion operator. We apply the error bounds to prove convergence of a multilevel Monte Carlo estimator for the expected value of the pathwise optimal controls. In addition we analyze the computational complexity of the multilevel estimator. We perform numerical experiments in two-dimensional space to confirm the convergence result and the complexity bound.

A posteriori error estimation for the steady Navier–Stokes equations in random domains

We consider finite element error approximations of the steady incompressible Navier–Stokes equations defined on a randomly perturbed domain, the perturbation being small. Introducing a random mapping, these equations are transformed into PDEs on a fixed reference domain with random coefficients. Under suitable assumptions on the random mapping and the input data, in particular the so-called small data assumption, we prove the well-posedness of the problem. We assume then that the mapping depends affinely on L independent random variables and adopt a perturbation approach expanding the solution with respect to a small parameter ε that controls the amount of randomness in the problem. We perform an a posteriori error analysis for the first order approximation error, namely the error between the exact (random) solution and the finite element approximation of the first term in the expansion with respect to ε. Numerical results are given to illustrate the theoretical results and the effectiveness of the error estimators.

A Priori Error Estimates for Three Dimensional Parabolic Optimal Control Problems with Pointwise Control

In this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise (Dirac-type) control in space on three-dimensional domains. The two-dimensional case was treated in [D. Leykekhman and B. Vexler, SIAM J. Numer. Anal., 51 (2013), pp. 2797--2821]; however, the three-dimensional case is technically much more involved. To approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we establish ${\mathcal O}(\sqrt{k}+h)$ order of convergence rate for the control in the $L^2$ norm. This result improves almost twice the previously known estimate in [W. Gong, M. Hinze, and Z. Zhou, SIAM J. Control Optim., 52 (2014), pp. 97--119] and in addition does not require any relationship between the time step $k$ and the mesh size $h$. The main technical tools are discrete maximal parabolic regularity results and the best approximation-type estimate for the finite element error in the $L^\infty(\Omega;L^2(I))$ norm.

A New Coupled Complex Boundary Method for Bioluminescence Tomography

AbstractIn this paper, we introduce and study a new method for solving inverse source problems, through a working model that arises in bioluminescence tomography (BLT). In the BLT problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter for trade off between the accuracy and stability of approximate solutions. The new method is based on a combination of the boundary condition and the boundary measurement in a parameter-dependent single complex Robin boundary condition, followed by the Tikhonov regularization. By properly adjusting the parameter in the Robin boundary condition, we achieve two important properties for our new method: first, the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy; second, the convergence order of the regularized solutions reaches one with respect to the noise level. Then, the finite element method is used to compute numerical solutions and a new finite element error estimate is derived for discrete solutions. These results improve related results found in the existing literature. Several numerical examples are provided to illustrate the theoretical results.

Adaptive finite element analysis for semilinear elliptic problems

In this paper, without using the uniform boundedness of finite element approximations, we derive both a priori and a posteriori finite element error estimates for a class of semilinear elliptic problems, based on which we analyze an adaptive finite element method to approximate the solutions of this kind of semilinear problems. In particular, we obtain the convergence rate and complexity of the adaptive finite element method.

Towards optimal finite element error estimates for the penalized Dirichlet problem in a domain with curved boundary

We consider the finite element approximation of an elliptic problem with homogeneous Dirichlet boundary conditions on a curved boundary and imposed using the penalty method. We establish optimal H1 error estimates with suitable assumptions on the penalty parameter ε as a function of the elements size h. Our focus is on establishing these results with least restrictive assumptions possible on this dependency.

A posteriori error analysis for finite element solution of one-dimensional elliptic differential equations using equidistributing meshes

The paper is concerned with the adaptive linear finite element solution of linear one-dimensional elliptic differential equations using equidistributing meshes. A strategy is developed for defining such meshes based on a residual-based a posteriori error estimate. The mesh and the finite element solution are determined by the coupled system of the finite element equation and the equidistribution relation (i.e., the mesh equation). An iterative algorithm is proposed for solving this system for the mesh and the finite element solution. The existence of an equidistributing mesh is proven for a given sufficiently large number of the points with help of a result on the continuous dependence of the finite element solution on the mesh, which is also established in the current work. Error bounds for the finite element solution are obtained for the equidistributing and quasi-equidistributing meshes. They show that adaptive meshes can lead to more accurate solutions than a uniform mesh and it is unnecessary to compute the equidistribution relation accurately for the equidistributing meshes. The departure from the equidistributing meshes has only a mild effect on the finite element error. Numerical examples are given to illustrate the convergence of the iterative algorithm and the theoretical findings.

Finite element error estimates for an optimal control problem governed by the Burgers equation

We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, a superlinear order of convergence for the control is obtained in the $$L^2$$L2-norm; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to $$h^{3/2}$$h3/2, extending the results in Rosch (Optim. Methods Softw. 21(1): 121---134, 2006). The theoretical findings are tested experimentally by means of numerical examples.

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(?+hl+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(?2+hl+1).

Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H1(Ω)‐norm, L2(Ω)‐norm and L2(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.