Galerkin Orthogonality Research Articles

Variational data assimilation with finite-element discretization for second-order parabolic interface equation

Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.

An Adaptive PML Finite Volume Algorithm for the Scattering by Periodic Gratings

In this paper, we studied the finite volume formulations for solving the diffraction grating problem that is truncated by the perfectly matched layer (PML) technique. Based on a reliable the a-posteriori error estimate, an adaptive PML finite volume method is discussed for the numerical approximation of the diffraction grating problem. The PML parameters are obtained numerically by sharp a-posteriori error estimates of the PML finite volume method such as the thickness of the layer and the medium property. It is worth mentioning in the a-posteriori error estimates that we derive the error representation formula and use a [Formula: see text]-orthogonality property of the residual similar to the Galerkin orthogonality used in the finite element method. Furthermore, the lower bound is established to demonstrate the efficiency of the a-posteriori error estimates. Numerical experiments are given to illustrate the accuracy and robustness of our adaptive algorithm.

A penalty-free Shifted Boundary Method of arbitrary order

We introduce and analyze a penalty-free formulation of the Shifted Boundary Method (SBM), inspired by the asymmetric version of the Nitsche method. We prove its stability and convergence for arbitrary order finite element interpolation spaces and we test its performance with a number of numerical experiments. Moreover, while the SBM was previously believed to be only asymptotically consistent (in the sense of Galerkin orthogonality), we prove here that it is indeed exactly consistent.

On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

Model reduction of the compressible Euler equations based on proper orthogonal decomposition (POD) and Galerkin orthogonality or least-squares residual minimization requires the selection of inner product spaces in which to perform projections and measure norms. The most popular choice is the vector-valued L2(Ω) inner product space. This choice, however, yields dimensionally-inconsistent reduced-order model (ROM) formulations which often lack robustness. In this work, we try to address this weakness by studying a set of dimensionally-consistent inner products with application to the compressible Euler equations. First, we demonstrate that non-dimensional inner products have a positive impact on both POD and Galerkin/least-squares ROMs. Second, we further demonstrate that physics-based inner products based on entropy principles result in drastically more accurate and robust ROM formulations than those based on non-dimensional L2(Ω) inner products. As test cases, we consider the following problems: the one-dimensional Sod shock tube, a two-dimensional Riemann problem, a two-dimensional Kelvin-Helmholtz instability, and two-dimensional homogeneous isotropic turbulence.

An MP-DWR method for h-adaptive finite element methods

In a dual-weighted residual method based on the finite element framework, the Galerkin orthogonality is an issue that prevents solving the dual equation in the same space as the one for the primal equation. In the literature, there have been two popular approaches to constructing a new space for the dual problem, i.e., refining mesh grids (h-approach) and raising the order of approximate polynomials (p-approach). In this paper, a novel approach is proposed for the purpose based on the multiple-precision technique, i.e., the construction of the new finite element space is based on the same configuration as the one for the primal equation, except for the precision in calculations. The feasibility of such a new approach is discussed in detail in the paper. In numerical experiments, the proposed approach can be realized conveniently with C++ template. Moreover, the new approach shows remarkable improvements in both efficiency and storage compared with the h-approach and the p-approach. It is worth mentioning that the performance of our approach is comparable with the one through a higher order interpolation (i-approach) in the literature. The combination of these two approaches is believed to further enhance the efficiency of the dual-weighted residual method.

Consistency results for the dual-wind discontinuous Galerkin method

This paper further analyzes the dual-wind discontinuous Galerkin (DWDG) method for approximating Poisson’s problem by directly examining the relationship between the Laplacian and the underlying discrete Laplacian. DWDG methods are derived from the DG differential calculus framework that defines discrete differential operators to replace continuous differential operators. We establish a priori error estimates for the DWDG approximation of Δ. Since the DWDG method does not satisfy a Galerkin orthogonality condition, we also explore the relationship between the DWDG approximation and the Ritz projection defined to satisfy an exact Galerkin orthogonality condition linked to the DWDG method. Numerical experiments are provided to validate the theoretical results.

An analysis of least-squares oversampled collocation methods for compactly perturbed boundary integral equations in two dimensions

In recent work (Maierhofer and Huybrechs, 2022, Adv. Comput. Math.), the authors showed that least-squares oversampling can improve the convergence properties of collocation methods for boundary integral equations involving operators of certain pseudo-differential form. The underlying principle is that the discrete method approximates a Bubnov–Galerkin method in a suitable sense. In the present work, we extend this analysis to the case when the integral operator is perturbed by a compact operator K which is continuous as a map on Sobolev spaces on the boundary, K:Hp→Hq for all p,q∈R.This study is complicated by the fact that both the test and trial functions in the discrete Bubnov–Galerkin orthogonality conditions are modified over the unperturbed setting. Our analysis guarantees that previous results concerning optimal convergence rates and sufficient rates of oversampling are preserved in the more general case. Indeed, for the first time, this analysis provides a complete explanation of the advantages of least-squares oversampled collocation for boundary integral formulations of the Laplace equation on arbitrary smooth Jordan curves in 2D. Our theoretical results are shown to be in very good agreement with numerical experiments.

Reliable a Posteriori Error Estimation for Cosserat Elasticity in 3D

A new a posteriori error estimate for Cosserat elasticity is proposed. Paper continues implementations of functional approach to error estimation for planar problems of classical and Cosserat elasticity. The proposed majorant is reliable regardless of some additional assumptions (for instance, the Galerkin orthogonality). This property is preserved independently of methods used for solving a problem, and the estimate is valid for accuracy control of any conforming approximate solution.

An accuracy analysis of Galerkin meshfree methods accounting for numerical integration

Presently, it has been widely recognized that numerical integration plays a crucial role on the solution accuracy of Galerkin meshfree methods due to the rational nature of meshfree shape functions and the misalignment of shape function supports with integration cells. Nonetheless, the corresponding underlying cause is still not quite clear. To address this issue, an accuracy analysis of Galerkin meshfree methods is presented in this study with a special focus on the contribution of numerical integration to the error estimates. One important fact is that the integration difficulty of meshfree methods leads to a loss of the Galerkin orthogonality condition which usually serves as a basis to develop error bounds. In order to enable the accuracy analysis, an error measure is proposed to evaluate the loss of Galerkin orthogonality condition, which is actually related to the order of integration consistency for Galerkin meshfree formulation. With the aid of this orthogonality error measure, both H1 and L2 error estimates are established for Galerkin meshfree methods. It turns out that these errors essentially consist of two parts, one comes from the standard interpolation error, and the other one attributes to the numerical integration error. In accordance with the proposed error estimates, it is evident that for the conventional Gauss integration schemes violating the integration consistency, the solution errors eventually are controlled by the integration error and optimal convergence cannot be achieved even with high order quadrature rules. On the other hand, for the consistent integration approaches, e.g., the stabilized conforming nodal integration and the more recent reproducing kernel gradient smoothing integration, the integration and interpolation errors have the same accuracy order and thus an optimal convergence of Galerkin meshfree methods is ensured. The proposed theoretical error estimates for Galerkin meshfree methods are well demonstrated by numerical results.

Consistent weak forms for meshfree methods: Full realization of [formula omitted]-refinement, [formula omitted]-refinement, and [formula omitted]-refinement in strong-type essential boundary condition enforcement

Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial due to the fact that field variables are not guaranteed to coincide with their coefficients at nodal locations. A common approach to overcome this issue is to strongly enforce the boundary conditions at these points by employing a technique to modify the approximation such that this is possible. However, with these methods, test and trial functions do not strictly satisfy the requirements of the conventional weak formulation of the problem, as the desired imposed values can actually deviate between nodes on the boundary. In this work, it is first shown that this inconsistency results in the loss of Galerkin orthogonality and the best approximation property, and correspondingly, failure to pass the patch test. It is also shown that this induces an O(h) error in the energy norm in the solution of second-order boundary value problems that is independent of the order of completeness in the approximation. As a result, this places a barrier on the global order of accuracy of Galerkin meshfree solutions to that of linear consistency. That is, with these methods, it is not possible to attain the higher order accuracy offered by meshfree approximations in the solution of boundary-value problems. To remedy this deficiency, two new weak forms are introduced that relax the requirements on the test and trial functions in the traditional weak formulation. These are employed in conjunction with strong enforcement of essential boundary conditions at nodes, and several benchmark problems are solved to demonstrate that optimal accuracy and convergence rates associated with the order of approximation can be restored using the proposed method. In other words, this approach allows p-refinement, and h-refinement with pth order rates with strong enforcement of boundary conditions beyond linear (p>1) for the first time. In addition, a new concept termed a-refinement is introduced, where improved accuracy is obtained by increasing the kernel measure in meshfree approximations, previously unavailable.

A tensor format for the generalized Hessenberg method for solving Sylvester tensor equations

In this paper, a general framework using tensor Krylov projection techniques is proposed for solving high order Sylvester tensor equations. After describing the tensor format of the generalized Hessenberg process, we combine the obtained different processes with a Galerkin orthogonality condition or with a minimal norm condition in order to derive the Hess−BTF and CMRH−BTF methods which are based on the tensor format of the Hessenberg process. In addition, we also recover the FOM−BTF and GMRES−BTF which are known methods based on the tensor format of the Arnoldi process. To accelerate the convergence or prevent a possible stagnation of the different obtained methods, we incorporate a weighting strategy based on the use of a weighted inner product instead of the classical one when building a basis for the tensor Krylov subspace. Numerical experiments are described in order to compare the new proposed methods that are Hess−BTF and CMRH−BTF with the known methods FOM−BTF and GMRES−BTF and to show the efficiency of the weighting strategy. Moreover, we utilize a flexible preconditioning framework for the unweighted and weighted forms of the proposed methods, and the flexible version is validated by satisfactory numerical results.

Pointwise error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Pointwise error analysis of the linear finite element approximation for $$-\,\Delta u + u = f$$ in $$\Omega $$, $$\partial _n u = \tau $$ on $$\partial \Omega $$, where $$\Omega $$ is a bounded smooth domain in $$\mathbb {R}^N$$, is presented. We establish $$O(h^2|\log h|)$$ and O(h) error bounds in the $$L^\infty $$- and $$W^{1,\infty }$$-norms respectively, by adopting the technique of regularized Green’s functions combined with local $$H^1$$- and $$L^2$$-estimates in dyadic annuli. Since the computational domain $$\Omega _h$$ is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy $$\Omega _h \ne \Omega $$. In particular, the so-called Galerkin orthogonality relation, utilized three times in the proof, does not exactly hold and involves domain perturbation terms (or boundary-skin terms), which need to be addressed carefully. A numerical example is provided to confirm the theoretical result.

Convergence and quasi-optimality of an adaptive continuous interior multi-penalty finite element method

An adaptive continuous interior multi-penalty finite element method (ACMPFEM) for symmetric second-order linear elliptic equations is considered. The proposed ACMPFEM is defined using a bilinear form which not only penalize the jumps of the first order normal derivatives across the element edges but also the jumps of all normal derivatives up to order p. In addition, compared with the analyses for adaptive finite element method, extra works are done to overcome the difficulties caused by the additional penalty terms. To be precise, the first difficulty is that the variational formulation of the ACMPFEM is not consistent with the continuous problem for weak solution u just in and the second difficulty lies in the proof of the optimality of our ACMPFEM. Furthermore, to prove the convergence and quasi-optimality of the ACMPFEM, an approximate Galerkin orthogonality in which the penalty term is regarded as a perturbation, and estimating the perturbation very carefully is used and the definition of our approximation class has to be involved with the penalty terms. Numerical tests are provided to verify the theoretical results and show advantages of the ACMPFEM.

The inf-sup condition and error estimates of the Nitsche method for evolutionary diffusion–advection-reaction equations

The Nitsche method is a method of "weak imposition" of the inhomogeneous Dirichlet boundary conditions for partial differential equations. This paper explains stability and convergence study of the Nitsche method applied to evolutionary diffusion-advection-reaction equations. We mainly discuss a general space semidiscrete scheme including not only the standard finite element method but also Isogeometric Analysis. Our method of analysis is a variational one that is a popular method for studying elliptic problems. The variational method enables us to obtain the best approximation property directly. Actually, results show that the scheme satisfies the inf-sup condition and Galerkin orthogonality. Consequently, the optimal order error estimates in some appropriate norms are proven under some regularity assumptions on the exact solution. We also consider a fully discretized scheme using the backward Euler method. Numerical example demonstrate the validity of those theoretical results.

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are $u_h$ in elements and $\hat{u}_h$ on inter-element edges. That is, we formulate our schemes without introducing the flux variable. Our schemes naturally satisfy the Galerkin orthogonality. The solution $u$ of the interface problem under consideration may not have a sufficient regularity, say $u|_{\Omega_1}\in H^2(\Omega_1)$ and $u|_{\Omega_2}\in H^2(\Omega_2)$, where $\Omega_1$ and $\Omega_2$ are subdomains of the whole domain $\Omega$ and $\Gamma=\partial\Omega_1\cap\partial\Omega_2$ implies the interface. We study the convergence, assuming $u|_{\Omega_1}\in H^{1+s}(\Omega_1)$ and $u|_{\Omega_2}\in H^{1+s}(\Omega_2)$ for some $s\in (1/2,1]$, where $H^{1+s}$ denotes the fractional order Sobolev space. Consequently, we succeed in deriving optimal order error estimates in an HDG norm and the $L^2$ norm. Numerical examples to validate our results are also presented.

Adjoint-Based, Superconvergent Galerkin Approximations of Linear Functionals

We propose a new technique for computing highly accurate approximations to linear functionals in terms of Galerkin approximations. We illustrate the technique on a simple model problem, namely, that of the approximation of J(u), where $$J(\cdot )$$ is a very smooth functional and u is the solution of a Poisson problem; we assume that the solution u and the solution of the adjoint problem are both very smooth. It is known that, if $$u_h$$ is the approximation given by the continuous Galerkin method with piecewise polynomials of degree $$k>0$$ , then, as a direct consequence of its property of Galerkin orthogonality, the functional $$J(u_h)$$ converges to J(u) with a rate of order $$h^{2k}$$ . We show how to define approximations to J(u), with a computational effort about twice of that of computing $$J(u_h)$$ , which converge with a rate of order $$h^{4k}$$ . The new technique combines the adjoint-recovery method for providing precise approximate functionals by Pierce and Giles (SIAM Rev 42(2):247–264, 2000), which was devised specifically for numerical approximations without a Galerkin orthogonality property, and the accuracy-enhancing convolution technique of Bramble and Schatz (Math Comput 31(137):94–111, 1977), which was devised specifically for numerical methods satisfying a Galerkin orthogonality property, that is, for finite element methods like, for example, continuous Galerkin, mixed, discontinuous Galerkin and the so-called hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for $$k=1,2,3$$ in one-space dimension and for $$k=1,2$$ in two-space dimensions, which show that $$J(u_h)$$ converges to J(u) with order $$h^{2k+1}$$ and that the new approximations converges with order $$h^{4k}$$ . The numerical experiments also indicate, for the p-version of the method, that the rate of exponential convergence of the new approximations is about twice that of $$J(u_h)$$ .

A Time-Space Adaptive Method for the Schrödinger Equation

AbstractIn this paper, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto finite elements in space. A truncated Galerkin orthogonality is used to obtain duality-based a posteriori error estimates that address the temporal and the spatial error separately. Based on this theory, a space-time adaptive solver for the Schrödinger equation is devised. An efficient matrix-free implementation of the differential operator, suited for spectral elements, is used to enable computations for realistic configurations. We demonstrate the performance of the algorithm for the example of matter-field interaction.

Unified a Priori Error Estimate and a Posteriori Error Estimate of CIP-FEM for Elliptic Equations

AbstractThis paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lowerH1+sweak regularity under consideration, where 0 ≤s≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Céa Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliaryC1finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

A Note on Multilevel Based Error Estimation

Abstract By employing the infinite multilevel representation of the residual, we derive computable bounds to estimate the distance of finite element approximations to the solution of the Poisson equation. If the finite element approximation is a Galerkin solution, the derived error estimator coincides with the standard element and edge based estimator. If Galerkin orthogonality is not satisfied, then the discrete residual additionally appears in terms of the BPX preconditioner. As a by-product of the present analysis, conditions are derived such that the hierarchical error estimation is reliable and efficient.

Adaptive Vertex-Centered Finite Volume Methods with Convergence Rates

We consider the vertex-centered finite volume method with first-order conforming ansatz functions. The adaptive mesh-refinement is driven by the local contributions of the weighted-residual error estimator. We prove that the adaptive algorithm leads to linear convergence with generically optimal algebraic rates for the error estimator and the sum of energy error plus data oscillations. While similar results have been derived for finite element methods and boundary element methods, the present work appears to be the first for adaptive finite volume methods, where the lack of the classical Galerkin orthogonality leads to new challenges.