Finite Element Error Estimates Research Articles

Finite element error estimates in non-energy norms for the two-dimensional scalar Signorini problem

This paper is concerned with error estimates for the piecewise linear finite element approximation of the two-dimensional scalar Signorini problem on a convex polygonal domain varOmega . Using a Céa-type lemma, a supercloseness result, and a non-standard duality argument, we prove W^{1,p}(varOmega )-, L^infty (varOmega )-, W^{1,infty }(varOmega )-, and H^{1/2}(partial varOmega )-error estimates under reasonable assumptions on the regularity of the exact solution and L^p(varOmega )-error estimates under comparatively mild assumptions on the involved contact sets. The obtained orders of convergence turn out to be optimal for problems with essentially bounded right-hand sides. Our results are accompanied by numerical experiments which confirm the theoretical findings.

Numerical analysis of sparse initial data identification for parabolic problems

In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results.

Error estimates for the finite element approximation of bilinear boundary control problems

In this article a special class of nonlinear optimal control problems involving a bilinear term in the boundary condition is studied. These kind of problems arise for instance in the identification of an unknown space-dependent Robin coefficient from a given measurement of the state, or when the Robin coefficient can be controlled in order to reach a desired state. Necessary and sufficient optimality conditions are derived and several discretization approaches for the numerical solution of the optimal control problem are investigated. Considered are both a full discretization and the postprocessing approach meaning that we compute an improved control by a pointwise evaluation of the first-order optimality condition. For both approaches finite element error estimates are shown and the validity of these results is confirmed by numerical experiments.

Explicit k-dependence for P k finite elements in W m,p error estimates: application to probabilistic laws for accuracy analysis

We derive an explicit k-dependence in error estimates for Lagrange finite elements. Two laws of probability are established to measure the relative accuracy between and finite elements, (), in terms of -norms. We further prove a weak asymptotic relation in between these probabilistic laws when difference goes to infinity. Moreover, as expected, one finds that finite element is surely more accurate than , for sufficiently small values of the mesh size h. Nevertheless, our results also highlight cases where is more likely accurate than , for a range of values of h. Hence, this approach brings a new perspective on how to compare two finite elements, which is not limited to the rate of convergence.

Optimal-order preconditioners for the Morse–Ingard equations

The Morse–Ingard equations of thermoacoustics Morse and Ingard (1986) are a system of coupled time-harmonic equations for the temperature and pressure of an excited gas. They form a critical aspect of modeling trace gas sensors. In this paper, we analyze a reformulation of the system that has a weaker coupling between the equations than the original form. We give a Gårding-type inequality for the system that leads to optimal-order asymptotic finite element error estimates. We also develop preconditioners for the coupled system. These are derived by writing the system as a 2 × 2 block system with pressure and temperature unknowns segregated into separate blocks and then using either the block diagonal or block lower triangular part of this matrix as a preconditioner. Consequently, the preconditioner requires inverting smaller, Helmholtz-like systems individually for the pressure and temperature. Rigorous eigenvalue bounds are given for the preconditioned system, and these are supported by numerical experiments.

Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures

Consider the scattering of a time-harmonic elastic plane wave by a periodic rigid surface. The elastic wave propagation is governed by the two-dimensional Navier equation. Based on a Dirichlet-to-Neumann (DtN) map, a transparent boundary condition (TBC) is introduced to reduce the scattering problem into a boundary value problem in a bounded domain. By using the finite element method, the discrete problem is considered, where the TBC is replaced by the truncated DtN map. A new duality argument is developed to derive the a posteriori error estimate, which contains both the finite element approximation error and the DtN truncation error. An a posteriori error estimate based adaptive finite element algorithm is developed to solve the elastic surface scattering problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

On subgrid multiscale stabilized finite element method for advection-diffusion-reaction equation with variable coefficients

In this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been introduced. Here subgrid scale approach along with algebraic approximation to the sub-scales has been chosen to stabilize the Galerkin finite element method. Both a priori and a posteriori finite element error estimates in L2 norm have been derived after introducing the stabilized variational form. An expression of the stabilization parameter has also been derived here. At last numerical experiments are presented to verify numerical performance of the stabilized method and the credibility of the theoretically derived expression of the stabilization parameter has been established numerically.

Finite element error estimates in $$L^2$$ for regularized discrete approximations to the obstacle problem

This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the $$L^2$$-norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly regularized problem. The underlying domain is only assumed to be convex and polygonally or polyhedrally bounded such that an application of pointwise error estimates results in a rate less than two in general. The main ingredient for proving the quasi-optimal estimates is the structural and commonly used assumption that the obstacle is inactive on the boundary of the domain. Then localization techniques are used to estimate the global $$L^2$$-error by a local error in the inner part of the domain, where higher regularity for the solution can be assumed, and a global error for the Ritz-projection of the solution, which can be estimated by standard techniques. We validate our results by numerical examples.

A FEM for an optimal control problem subject to the fractional Laplace equation

We study the numerical approximation of linear–quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed. Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

Abstract The purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order 𝒪 ( H s Log ^ ( H h ) s 2 ) {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and 𝒪 ( H 2 ⁢ s Log ^ ( H h ) s ) {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the L ∞ ⁢ ( H 1 ) {L^{\infty}(H^{1})} -norm and L ∞ ⁢ ( L 2 ) {L^{\infty}(L^{2})} -norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

We consider an optimal control problem governed by a one-dimensional elliptic equation that involves univariate functions of bounded variation as controls. For the discretization of the state equation we use linear finite elements and for the control discretization we analyze two strategies. First, we use variational discretization of the control and show that the $ L^2 $- and $ L^\infty $-error for the state and the adjoint state are of order $ {\mathcal O}(h^2) $ and that the $ L^1 $-error of the control behaves like $ {\mathcal O}(h^2) $, too. These results rely on a structural assumption that implies that the optimal control of the original problem is piecewise constant and that the adjoint state has nonvanishing first derivative at the jump points of the control. If, second, piecewise constant control discretization is used, we obtain $ L^2 $-error estimates of order $ \mathcal{O}(h) $ for the state and $ W^{1, \infty} $-error estimates of order $ \mathcal{O}(h) $ for the adjoint state. Under the same structural assumption as before we derive an $ L^1 $-error estimate of order $ \mathcal{O}(h) $ for the control. We discuss optimization algorithms and provide numerical results for both discretization schemes indicating that the error estimates are optimal.

Finite Element Error Estimates for Normal Derivatives on Boundary Concentrated Meshes

This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustr...

Logarithm cannot be removed in maximum norm error estimates for linear finite elements in 3D

For linear finite element discretizations of the Laplace equation in three dimensions, we give an example of a tetrahedral mesh in the cubic domain for which the logarithmic factor cannot be removed from the standard upper bounds on the error in the maximum norm.

A posteriori error analysis for the scattering by obstacles in a homogeneous chiral environment

In this paper we consider the scattering of time-harmonic electromagnetic wave propagation in a homogeneous chiral environment by obstacles. The model is simplified to a two-dimensional scattering problem, and is formulated as a boundary value problem in a bounded domain by introducing the nonlocal boundary conditions associated with Dirichlet-to-Neumann operators. An a posteriori error estimate is established when the truncation of the nonlocal boundary operators takes place. The crucial part of the error analysis is to develop a duality argument and use the Bohren decomposition of the electromagnetic fields. The a posteriori error estimate consists of two parts, finite element approximation error and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter. Numerical experiments are also presented to show the robustness and effectiveness of our numerical algorithm.

A priori error estimates of a combined mixed finite element and local discontinuous Galerkin method for an incompressible miscible displacement problem

A numerical approximation for a kind of incompressible miscible displacement problems in high dimension in porous media is studied. Mixed finite element method is applied to the flow equation, and the transport one is solved by the local discontinuous Galerkin method (LDG). Based on interpolation projection properties and the induction hypothesis, a priori hp error estimates are obtained. Numerical results are presented, which verify the theoretical results.

Semidiscrete Finite Element Analysis of Time Fractional Parabolic Problems: A Unified Approach

In this paper, we consider the numerical approximation of time-fractional parabolic problems involving Caputo derivatives in time of order $\alpha$, $0< \alpha<1$. We derive optimal error estimates for semidiscrete Galerkin finite element (FE) type approximations for problems with smooth and nonsmooth initial data. Our analysis relies on energy arguments and exploits the properties of the inverse of the associated elliptic operator. We present the analysis in a general setting so that it is easily applicable to various spatial approximations such as conforming and nonconforming FE methods (FEMs) and FEM on nonconvex domains. The finite element approximation in mixed form is also presented and new error estimates are established for smooth and nonsmooth initial data. Finally, an extension of our analysis to a multiterm time-fractional model is discussed.

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.

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.

The minimal angle condition for quadrilateral finite elements of arbitrary degree

We study W1,p Lagrange interpolation error estimates for general quadrilateralQk finite elements with k≥2. For the most standard case of p=2 it turns out that the constant C involved in the error estimate can be bounded in terms of the minimal interior angle of the quadrilateral. Moreover, the same holds for any p in the range 1≤p<3. On the other hand, for 3≤p we show that C also depends on the maximal interior angle. We provide some counterexamples showing that our results are sharp.

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.