Convergence Of Finite Element Approximations Research Articles

Optimal convergence of finite element approximation to an optimization problem with PDE constraint* *Wei Gong was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB 41000000), the National Key Basic Research Program (Grant No. 2018YFB0704304) and the National Natural Science Foundation of China (Grant No. 12071468, 11671391). Zhaojie Zhou was supported by the National

We study in this paper the optimal convergence of finite element approximation to an optimization problem with PDE constraint. Specifically, we consider an elliptic distributed optimal control problem without control constraints, which can also be viewed as a regularized inverse source problem. The main contributions are two-fold. First, we derive a priori and a posteriori error estimates for the optimization problems, under an appropriately chosen norm that allows us to establish an isomorphism between the solution space and its dual. These results yield error estimates with explicit dependence on the regularization parameter α so that the constants appeared in the derivation are independent of α. Second, we prove the contraction property and rate optimality for the adaptive algorithm with respect to the error estimator and solution errors between the adaptive finite element solutions and the continuous solutions. Extensive numerical experiments are presented that confirm our theoretical results.

On the approximation of dispersive electromagnetic eigenvalue problems in two dimensions

Abstract We consider time-harmonic electromagnetic wave equations in composites of a dispersive material surrounded by a classical material. In certain frequency ranges this leads to sign-changing permittivity and/or permeability. Previously meshing rules were reported, which guarantee the convergence of finite element approximations to the related scalar source problems. Here we generalize these results to the electromagnetic two-dimensional vectorial equations and the related holomorphic eigenvalue problems. Different than for the analysis on the continuous level, we require an assumption on both contrasts of the permittivity and the permeability similar to the three-dimensional case. We confirm our theoretical results with computational studies.

Regularity analyses and approximation of nonlocal variational equality and inequality problems

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These types of operators are used to model anomalous diffusion and, for a special choice of the integral kernels, reduce to the fractional Laplace operator on a bounded domain. By means of a nonlocal vector calculus we recast the problems in a weak form, leading to corresponding nonlocal variational equality and inequality problems. We prove optimal regularity results for both problems, including a higher regularity of the solution and the Lagrange multiplier. Based on the regularity results, we analyze the convergence of finite element approximations for a linear problem and illustrate the theoretical findings by numerical results.

A Regularization Approach for an Inverse Source Problem in Elliptic Systems from Single Cauchy Data

In this article, we investigate the problem of identifying the source term f in the elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data (j, g) with noise level In this context, the diffusion matrix Q is given. A variational method of Tikhonov-type regularization with specific misfit term of Kohn–Vogelius-type and quadratic stabilizing penalty term is suggested to tackle this linear inverse problem. The method also appears as a variant of the Lavrentiev regularization. For the occurring linear inverse problem in infinite dimensional Hilbert spaces, convergence and rate results can be found from the general theory of classical Tikhonov and Lavrentiev regularization. Using the variational discretization concept, where the PDE is discretized with piecewise linear and continuous finite elements, we show the convergence of finite element approximations to a sought source function. Moreover, we derive an error bound and corresponding convergence rates provided a suitable range-type source condition is satisfied. For the numerical solution, we propose a conjugate gradient method. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.

Convergence of finite element approximation for electrical impedance tomography with the complete electrode model

This paper discusses the convergence of the finite element approximation for the forward problem of electrical impedance tomography with the complete electrode model. When the domain is polygonal and conforming finite element method is employed with linear triangle or tetrahedron elements, we prove that the estimated voltages on the electrodes converge to the true ones up to an additive constant, as the sizes of elements in the underlying mesh all tend to zero.

An anisotropic finite element method on polyhedral domains: Interpolation error analysis

On a polyhedral domain Ω ⊂ R 3 \Omega \subset \mathbb R^3 , consider the Poisson equation with the Dirichlet boundary condition. For singular solutions from the non-smoothness of the domain boundary, we propose new anisotropic mesh refinement algorithms to improve the convergence of finite element approximation. The proposed algorithm is simple, explicit, and requires less geometric conditions on the mesh and on the domain. Then, we develop interpolation error estimates in suitable weighted spaces for the anisotropic mesh, especially for the tetrahedra violating the maximum angle condition. These estimates can be used to design optimal finite element methods approximating singular solutions. We report numerical test results to validate the method.

Analysis and approximation of a nonlocal obstacle problem

An obstacle problem for a nonlocal operator is considered; the operator is a nonlocal integral analogue of the Laplacian operator and, as a special case, reduces to the fractional Laplacian. In the analysis of classical obstacle problems for the Laplacian, the obstacle is taken to be a smooth function. For the nonlocal obstacle problem considered here, obstacles are allowed to have jump discontinuities. We cast the nonlocal obstacle problem as a minimization problem wherein the solution is constrained to lie above the obstacle. We prove the existence and uniqueness of a solution in an appropriate function space. Then, the well posedness and convergence of finite element approximations are demonstrated. The results of numerical experiments are provided that illustrate the theoretical results and the differences between solutions of local, i.e., partial differential equation, and nonlocal obstacle problems.

Finite element approximation and iterative method solution of elliptic control problem with constraints to gradient of state

An optimal control problem with distributed control in the right-hand side of Poisson equation is considered. Pointwise constraints on the gradient of state and control are imposed in this problem. The convergence of finite element approximation for this problem is proved. Discrete saddle point problem is constructed and preconditioned Uzawa-type iterative algorithm for its solution is investigated.

Weak Convergence of Finite Element Approximations of Linear Stochastic Evolution Equations with Additive Lévy Noise

We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive Levy noise. We first derive a representation formula for the error which we then apply to study space-time discretizations of the stochastic heat equation, a Volterra-type integro-differential equation, and the wave equation as examples. For twice continuously differentiable test functions with bounded second derivative (with an additional condition on the second derivative for the wave equation) the weak rate of convergence is found to be twice the strong rate. The results extend earlier work by two of the authors, as we consider general square-integrable infinite-dimensional Levy processes and do not require boundedness of the test functions and their first derivative. Furthermore, the present framework is applicable to both hyperbolic and parabolic equations, and even to stochastic Volterra integro-differential equations.

Some theoretical and computational aspects of single‐crystal strain‐gradient plasticity

AbstractVariational formulations are constructed for rate‐independent problems in single‐crystal strain‐gradient plasticity. The framework makes use of the flow rule expressed in terms of a dissipation function. The formulation extends to the finite‐deformation context earlier work on this problem. Provision is made for energetic and dissipative microstresses, and a range of defect energies is accounted for. The minimization problem corresponding to the time‐discrete formulation is derived. Two special cases are then treated: first, for the small‐strain problem with energetic microstresses, results on well‐posedness, convergence of finite element approximations, the associated algorithms, and computational examples, are discussed. Secondly, recent computational work on the the related large‐deformation viscoplastic problem, with both energetic and dissipative effects, is presented and discussed for problems involving ensembles of grains.

Stokes equations with penalised slip boundary conditions

We consider the finite-element approximation of Stokes equations with slip boundary conditions imposed with the penalty method. In the case of a smooth curved boundary, our numerical results suggest that curved finite elements, regularised normal vectors or reduced integration techniques can be used to avoid a Babuska’s-type paradox and ensure the convergence of finite-element approximations to the exact solution. Convergence orders with these remedies are also compared.

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation.

Stabilisation yields strong convergence of macroscopic magnetisation vectors for micromagnetics without exchange energy

The convexified Landau–Lifshitz minimisation problem in micromagnetics leads to a degenerate variational problem. Therefore strong convergence of finite element approximations cannot be expected in general. This paper introduces a stabilized finite element discretization which allows for surprising the strong convergence of the discrete magnetisation fields with reduced convergence order for a uniaxial model problem. This yields a convergent method for the approximation of the Young measure which characterises the enforced microstructure for the generalized solution of the non-relaxed Landau–Lifshitz problem.

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise

A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An error representation formula is found in an abstract setting based on the semigroup formulation of stochastic evolution equations. This is then applied to the stochastic heat, linearized Cahn-Hilliard, and wave equations. In all cases it is found that the rate of weak convergence is twice the rate of strong convergence, sometimes up to a logarithmic factor, under the same or, essentially the same, regularity requirements.

Stability of semilinear elliptic optimal control problems with pointwise state constraints

A semi-linear elliptic control problems with distributed control and pointwise inequality constraints on the control and the state is considered. The general optimization problem is perturbed by a certain class of perturbations, and we establish convergence of local solutions of the perturbed problems to a local solution of the unperturbed optimal control problem. This class of perturbations include finite element discretization as well as data perturbation such that the theory implies convergence of finite element approximation and stability w.r.t. noisy data.

Finite element exterior calculus: from Hodge theory to numerical stability

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodge-theoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.

Convergence analysis and computational testing of the finite element discretization of the Navier–Stokes alpha model

AbstractThis report performs a complete analysis of convergence and rates of convergence of finite element approximations of the Navier–Stokes‐α (NS‐α) regularization of the NSE, under a zero‐divergence constraint on the velocity, to the true solution of the NSE. Convergence of the discrete NS‐α approximate velocity to the true Navier–Stokes velocity is proved and rates of convergence derived, under no‐slip boundary conditions. Generalization of the results herein to periodic boundary conditions is evident. Two‐dimensional experiments are performed, verifying convergence and predicted rates of convergence. It is shown that the NS‐α‐FE solutions converge at the theoretical limit of O(h2) when choosing α = h, in the H1 norm. Furthermore, in the case of flow over a step the NS‐α model is shown to resolve vortex separation in the recirculation zone. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

An augmented Lagrangian approach to the numerical solution of a non-smooth eigenvalue problem

In this article, we address the numerical solution of a non-smooth eigenvalue problem, which has implications in plasticity theory and image processing. The smallest eigenvalue of the non-smooth operator under consideration is shown to be the same for all bounded, sufficiently smooth, domains in two space dimensions. Piecewise linear finite elements are used for the discretization of eigenfunctions and eigenvalues. An augmented Lagrangian method is proposed for the computation of the minima of the associated non-convex optimization problem. The convergence of finite element approximations of generalized eigenpairs is investigated. Numerical solutions are presented for the first eigenvalue and eigenfunction. For non-simply connected domains, the augmented Lagrangian method also captures larger eigenvalues as local minima. Bifurcation between the first and second eigenvalues is investigated numerically.

Topological Derivatives for Semilinear Elliptic Equations

Topological Derivatives for Semilinear Elliptic EquationsThe form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in theL∞norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

Strong convergence for large bodies in micromagnetics

AbstractThe convexified Landau‐Lifshitz minimisation problem in micromagnetics leads to a degenerate variational problem. Therefore strong convergence of finite element approximations cannot be expected in general. This paper introduces a stabilised finite element discretisation which allows for the strong convergence of the discrete magnetisation fields with reduced convergence order for a uniaxial model problem. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)