Error Analysis For Discretization Research Articles

Equivalence of local-best and global-best approximations in H(curl)

We derive results on equivalence of piecewise polynomial approximations of a given function in the Sobolev space $${\varvec{H}}{\mathrm{(curl)}}$$ . We namely show that the global-best approximation of a given $${\varvec{H}}{\mathrm{(curl)}}$$ function in a $${\varvec{H}}{\mathrm{(curl)}}$$ -conforming piecewise polynomial space imposing the continuity of the tangential trace can be bounded above and below by the Hilbertian sum of the respective local approximations from the elementwise spaces without any inter-element continuity requirement. In other words, the approximation of a $${\varvec{H}}{\mathrm{(curl)}}$$ function by tangential-trace-continuous and discontinuous piecewise polynomials has comparable precision. We consider approximations of the curl of the target function in the $${\varvec{L}} ^2$$ -norm, as well as approximations of the target function in the $${\varvec{L}} ^2$$ -norm with a constraint on the curl; in the latter case, the constraint is removed in the local approximations. These best-approximation localizations hold under the minimal $${\varvec{H}}{\mathrm{(curl)}}$$ regularity, on arbitrary shape-regular tetrahedral meshes, and include imposition of conditions on a part of the boundary. They extend to the $${\varvec{H}}{\mathrm{(curl)}}$$ context some recent results from the $$H^1$$ and $${\varvec{H}}{\mathrm{(div)}}$$ spaces and have direct applications to a priori and a posteriori error analysis of numerical discretizations related to the $${\varvec{H}}{\mathrm{(curl)}}$$ space, namely Maxwell’s equations.

Error analysis for space discretizations of quasilinear wave-type equations

Abstract In this paper we study space discretizations of a general class of first- and second-order quasilinear wave-type problems. We present a rigorous error analysis based on a combination of inverse estimates with semigroup theory for nonautonomous linear Cauchy problems. Moreover, we provide refined results for the special case that the nonlinearities are local in space. As applications of these general results we derive novel error estimates for two prominent examples from nonlinear physics: the Westervelt equation and the Maxwell equations with Kerr nonlinearity. We conclude with a numerical example to illustrate our theoretical findings.

A priori error analysis for a mixed VEM discretization of the spectral problem for the Laplacian operator

The aim of the present work is to derive error estimates for the Laplace eigenvalue problem in mixed form, implementing a virtual element method. With the aid of the theory for non-compact operators, we prove that the proposed method is spurious free and convergent. Optimal order of convergence for the eigenvalues and eigenfunctions are derived. Finally, we report numerical tests to confirm the theoretical results together with a rigorous computational analysis of the effects of the stabilization parameter, inherent for the virtual element methods, in the computation of the spectrum.

Optimal Energy Conserving Local Discontinuous Galerkin Methods for Elastodynamics: Semi and Fully Discrete Error Analysis

We present an arbitrary high-order local discontinuous Galerkin (LDG) method with alternating fluxes for solving linear elastodynamics problems in isotropic media. Both the semi-discrete analysis and fully discrete analysis for a leap-frog LDG method are given to show that the proposed method simultaneously enjoys the energy conserving property and optimal convergence rates in both the displacement and stress, when the tensor product polynomials of the degree k are used on Cartesian meshes. Numerical experiments demonstrate that the proposed method has several advantages including the exact energy conservation, slow-growing errors in long time simulation, and subtle dependence on the first Lame parameter $$\lambda $$ .

The Performance of a Gradient-Based Method to Estimate the Discretization Error in Computational Fluid Dynamics

Although the grid convergence index is a widely used for the estimation of discretization error in computational fluid dynamics, it still has some problems. These problems are mainly rooted in the usage of the order of a convergence variable within the model which is a fundamental variable that the model is built upon. To improve the model, a new perspective must be taken. By analyzing the behavior of the gradient within simulation data, a gradient-based model was created. The performance of this model is tested on its accuracy, precision, and how it will affect a computational time of a simulation. The testing is conducted on a dataset of 36 simulated variables, simulated using the method of manufactured solutions, with an average of 26.5 meshes/case. The result shows the new gradient based method is more accurate and more precise then the grid convergence index(GCI). This allows for the usage of a coarser mesh for its analysis, thus it has the potential to reduce the overall computational by at least by 25% and also makes the discretization error analysis more available for general usage.

H-adaptive topology optimization considering variations of material properties and energy error density recovery

PurposeThe purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity.Design/methodology/approachThe new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level.FindingsTwo-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure.Originality/valueThis paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

Finite element discretization of semilinear acoustic wave equations with kinetic boundary conditions

We consider isoparametric finite element discretizations of semilinear acoustic wave equations with kinetic boundary conditions and derive a corresponding error bound as our main result. The difficulty is that such problems are stated on domains with curved boundaries and this renders the discretizations nonconforming. Our approach is to provide a unified error analysis for nonconforming space discretizations for semilinear wave equations. In particular, we introduce a general, abstract framework for nonconforming space discretizations in which we derive a-priori error bounds in terms of interpolation, data, and conformity errors. The theory applies to a large class of problems and discretizations that fit into the abstract framework. The error bound for wave equations with kinetic boundary conditions is obtained from the general theory by inserting known interpolation and geometric error bounds into the abstract error result of the unified error analysis.

Space–time adaptive finite elements for nonlocal parabolic variational inequalities

This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic variational inequalities associated to the fractional Laplacian we obtain a priori and a posteriori error estimates and study the resulting space–time adaptive mesh-refinement procedures. Particular emphasis is placed on mixed formulations, which include the contact forces as a Lagrange multiplier. Corresponding results are presented for elliptic problems. Our numerical experiments for 2-dimensional model problems confirm the theoretical results: They indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of space–time adaptive, as well as uniform and graded discretizations.

Heterogeneous Multiscale Method for Maxwell's Equations

We present a finite element heterogeneous multiscale method for time-dependent Maxwell's equations in first-order formulation in highly oscillatory materials using Nédélec's edge elements. Based on a uniform approach for the error analysis of nonconforming space discretizations [D. Hipp, M. Hochbruck, and C. Stohrer, IMA J. Numer. Anal., 39 (2019), pp. 1206--1245], we prove an error bound for the semidiscrete scheme. We further present error bounds for the fully discrete scheme, where we consider time discretization using algebraically stable Runge--Kutta methods, the Crank--Nicolson method, and the leapfrog method. These error bounds are confirmed by numerical experiments.

Unified error analysis for nonconforming space discretizations of wave-type equations

AbstractThis paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

Error analysis of non inf-sup stable discretizations of the time-dependent Navier–Stokes equations with local projection stabilization

Abstract This paper studies non inf-sup stable finite element approximations to the evolutionary Navier–Stokes equations. Several local projection stabilization (LPS) methods corresponding to different stabilization terms are analyzed, thereby separately studying the effects of the different stabilization terms. Error estimates are derived in which the constants are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure finite elements of degree $l$, it will be proved that the velocity error in $L^\infty (0,T;L^2(\varOmega ))$ decays with rate $l+1/2$ in the case that $\nu \le h$, with $\nu$ being the dimensionless viscosity and $h$ being the mesh width. In the analysis of another method it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies confirm the analytical results.

Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection , and we study the approximation of the probability distribution of the tra-jectories: test functions are regular, but depend on the values of the process on the interval [0, T ]. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Ito map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Ito map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space

Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin–Petrov time discretization schemes that is combined with a mixed finite element approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established. For this, the Banach–Nečas–Babuška theorem is applied in a non-standard way. Error estimates with explicit rates of convergence are proved for the scalar and vector-valued variable. An optimal order estimate in space and time is proved by duality techniques for the scalar variable. The convergence rates are analyzed and illustrated by numerical experiments, also on stochastically perturbed meshes.

A posteriori error analysis of semilinear parabolic interface problems using elliptic reconstruction

In this article, a posteriori error analysis for space-time discretizations of semilinear parabolic interface problems in a bounded convex domain in is presented and analyzed. In time discretizations both the backward Euler and the Crank-Nicolson approximations are considered whereas in space we have considered the standard piecewise linear finite elements. A posteriori error estimates of optimal order in time and almost optimal order in space are derived in the -norm. The main technical tools used are the energy argument combined with the elliptic reconstruction technique. The forcing term is assumed to satisfy the Lipschitz condition.

A new nonconforming finite element method for convection dominated diffusion-reaction equations

In this article, we introduce a non-conforming streamline diffusion finite element method for the approximation of convection dominated diffusion problems. The proposed non-conforming scheme is a hybrid type in which $$P_{1}$$ -nonconforming polynomials and $$P_{2}$$ -conforming polynomials on triangles are used. Computational advantages of the newly constructed element with rigorous error analysis for the streamline diffusion discretization is presented. Moreover, numerical experiments are conducted to demonstrate the efficiency of the proposed scheme and to validate the theoretical error estimates.

Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces

Convergence results are shown for full discretizations of quasilinear parabolic partial differential equations on evolving surfaces. As a semidiscretization in space the evolving surface finite element method is considered, using a regularity result of a generalized Ritz map, optimal order error estimates for the spatial discretization is shown. Combining this with the stability results for Runge–Kutta and backward differentiation formulae time integrators, we obtain convergence results for the fully discrete problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1200–1231, 2016

A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order μ ź (0, 1) with variable coefficients. For the spatial discretization, we apply the standard continuous Galerkin method of total degree ≤ 1 on each spatial mesh elements. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval (0, T) and a spatial domain Ω, our analysis suggest that the error in L2(0,T),L2(Ω)$L^{2}\left ((0,T),L^{2}({\Omega })\right )$-norm is O(k2źμ2+h2)$O(k^{2-\frac {\mu }{2}}+h^{2})$ (that is, short by order μ2$\frac {\mu }{2}$ from being optimal in time) where k denotes the maximum time step, and h is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal O(k2 + h2) error bound in the stronger Lź(0,T),L2(Ω)$L^{\infty }\left ((0,T),L^{2}({\Omega })\right )$-norm. Variable time steps are used to compensate the singularity of the continuous solution near t = 0.

A Higher Order Finite Element Method for Partial Differential Equations on Surfaces

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations

This paper is concerned with the discretization error analysis of semilinear Neumann boundary control problems in polygonal domains with pointwise inequality constraints on the control. The approximations of the control are piecewise constant functions. The state and adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of locally optimal controls of the continuous optimal control problem are constructed by the projection of the respective discrete adjoint state. Although the quality of the approximations is in general affected by corner singularities a convergence order of $h^2|\ln h|^{3/2}$ is proven for domains with interior angles smaller than $2\pi/3$ using quasi-uniform meshes. For larger interior angles mesh grading techniques are used to get the same order of convergence.

Analysis of trace finite element methods for surface partial differential equations

In this paper we consider two variants of a trace finite element method for solving elliptic partial differential equations on a stationary smooth manifold Γ . A discretization error analysis for both methods in one general framework is presented. Higher-order finite elements are treated and rather general numerical approximations Γh of the manifold Γ are allowed. Optimal-order discretization error bounds are derived. Furthermore, the conditioning of the stiffness matrices is studied. It is proved that for one of these two variants the corresponding scaled stiffness matrix has a condition number ∼ h−2, independently of how Γh intersects the outer triangulation.