Second-order Elliptic Problems Research Articles

A Penalty-Free and Essentially Stabilization-Free DG Method for Convection-Dominated Second-Order Elliptic Problems

A Penalty-Free and Essentially Stabilization-Free DG Method for Convection-Dominated Second-Order Elliptic Problems

A unified immersed finite element error analysis for one-dimensional interface problems

It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.

On the approximation of bi-Lipschitz maps by invertible neural networks

Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in applications. The universal approximation properties of INNs have been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems.

An analysis of nonconforming virtual element methods on polytopal meshes with small faces

In this paper, we analyze nonconforming virtual element methods on polytopal meshes with small faces for the second-order elliptic problem. We propose new stability forms for 2D and 3D nonconforming virtual element methods. For the 2D case, the stability form is defined by the sum of an inner product of approximate tangential derivatives and a weighed [Formula: see text]-inner product of certain projections on the mesh element boundaries. For the 3D case, the stability form is defined by a weighted [Formula: see text]-inner product on the mesh element boundaries. We prove the optimal convergence of the nonconforming virtual element methods equipped with such stability forms. Finally, several numerical experiments are presented to verify our analysis and compare the performance of the proposed stability forms with the standard stability form [B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016) 879–904].

Hp-Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

Efficient implementation of the hybridized Raviart-Thomas mixed method by converting flux subspaces into stabilizations

<abstract><p>We show how to reduce the computational time of the practical implementation of the Raviart-Thomas mixed method for second-order elliptic problems. The implementation takes advantage of a recent result which states that certain local subspaces of the vector unknown can be eliminated from the equations by transforming them into stabilization functions; see the paper published online in JJIAM on August 10, 2023. We describe in detail the new implementation (in MATLAB and a laptop with Intel(R) Core (TM) i7-8700 processor which has six cores and hyperthreading) and present numerical results showing 10 to 20% reduction in the computational time for the Raviart-Thomas method of index $ k $, with $ k $ ranging from 1 to 20, applied to a model problem.</p></abstract>

Conforming VEM for general second-order elliptic problems with rough data on polygonal meshes and its application to a Poisson inverse source problem

This paper focuses on the analysis of conforming virtual element methods (VEMs) for general second-order linear elliptic problems with rough source terms and applies it to a Poisson inverse source problem with rough measurements. For the forward problem, when the source term belongs to [Formula: see text], the right-hand side for the discrete approximation defined through polynomial projections is not meaningful even for standard conforming VEM. The modified discrete scheme in this paper introduces a novel companion operator in the context of conforming VEM and allows data in [Formula: see text]. This paper has three main contributions. The first contribution is the design of a conforming companion operator [Formula: see text] from the conforming virtual element space to the Sobolev space [Formula: see text], a modified virtual element scheme, and the a priori error estimate for the Poisson problem in the best-approximation form without data oscillations. The second contribution is the extension of the a priori analysis to general second-order elliptic problems with source term in [Formula: see text]. The third contribution is an application of the companion operator in a Poisson inverse source problem when the measurements belong to [Formula: see text]. Tikhonov’s regularization technique regularizes the ill-posed inverse problem, and the conforming VEM approximates the regularized problem given a finite measurement data. The inverse problem is also discretized using the conforming VEM and error estimates are established. Numerical tests on different polygonal meshes for general second-order problems, and for a Poisson inverse source problem with finite measurement data verify the theoretical results.

Rates of robust superlinear convergence of preconditioned Krylov methods for elliptic FEM problems

This paper considers the iterative solution of finite element discretizations of second-order elliptic boundary value problems. Mesh independent estimations are given for the rate of superlinear convergence of preconditioned Krylov methods, involving the connection between the convergence rate and the Lebesgue exponent of the data. Numerical examples demonstrate the theoretical results.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints

We investigate discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints on general polygonal domains. We show that discontinuous Galerkin methods for general second-order elliptic boundary value problems can be used to solve the elliptic optimal control problems with pointwise state constraints. We establish concrete error estimates and numerical experiments are shown to support the theoretical results.

Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems

Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems

Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems

Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems

A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids

A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids

Families of hybridized interior penalty discontinuous Galerkin methods for locally degenerate advection-diffusion-reaction problems

We analyze families of primal high-order hybridized discontinuous Galerkin (HDG) methods for solving degenerate second-order elliptic problems. One major problem regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the whole domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptical in the rest. This problem is thus quite delicate to solve since the exact solution can be discontinuous at interfaces separating the elliptic and hyperbolic parts. The proposed hybridized interior penalty DG (H-IP) method is developed in a unified and compact fashion. It can handle pure-diffusive or -advective regimes as well as intermediate regimes that combine these two mechanisms for a wide range of Péclet numbers, including the tricky case of local evanescent diffusivity. To this end, an adaptive stabilization strategy based on the addition of jump-penalty terms is considered. An upwind-based scheme using a Lax–Friedrichs correction is favored for the hyperbolic region, and a Scharfetter–Gummel-based technique is preferred for the elliptic one. The well-posedness of the H-IP method is also briefly discussed by analyzing the strong consistency and the discrete coercivity condition in a self-adaptive energy-norm that is regime dependent. Extensive numerical experiments are performed to verify the model’s robustness for all the abovementioned regimes.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

A study of high-order immersed finite element spaces by pointwise interface conditions on curved interfaces

We present a study of p -th degree immersed finite element (IFE) spaces based on pointwise interface conditions that lead to optimal convergence rates for typical second-order elliptic interface problems on interface-independent meshes. We study the existence and approximation properties of IFE spaces spanned by functions that satisfy the extended interface conditions in a pointwise manner on general curved interfaces. We present computational results to demonstrate the approximation capabilities of the proposed IFE spaces.

Error analysis of Petrov-Galerkin immersed finite element methods

This paper designs and analyzes a new and stable Petrov–Galerkin (PG) immersed finite element method (IFEM) for the second-order elliptic interface problems by introducing stabilization terms based on the classical PG-IFEM, which lacks the local positivity. The Petrov–Galerkin immersed finite element method uses the immersed finite element functions for the trial space and the standard finite element functions for the test space. Both the a priori and a posteriori error estimates of the method are analyzed in this paper. We prove the continuity and inf-sup condition and the a priori error estimate of the energy norm. The proposed a posteriori error estimator is proved to be both reliable and efficient, with both reliability and efficiency constants independent of the location of the interface. Extensive numerical results confirm the numerical scheme’s optimal convergence and indicate the robustness with respect to the interface-mesh intersection and the coefficient contrast, despite the robustness of the inf-sup constant with respect to the interface-mesh intersection has yet been theoretically proved.

Isoparametric numerical integration on enriched 4D simplicial elements

The enriched finite element method is a natural improvement of the classical finite element method for solving complicated engineering problems. It is an essential tool for successful treating difficulties, singularities, non-Lipschitz internal boundaries, curved boundaries with complex geometry, crack propagations, etc. This paper deals with an enrichment of the four-dimensional isoparametric quadratic Lagrangian finite element. The paper discusses a new approach in which bubble functions are replaced with specific cubic ones in the enrichment process. The new method is applied to simplicial elements in four-dimensional Euclidean space, where lumping of the mass matrix is obtained. Error estimates in numerical integration and the interpolation by the new elements are proved. Applications of the quadrature rules for solving elliptic eigenvalue problems are demonstrated. A new operator that maps tesseract onto a four-dimensional ball is defined. The new quadrature rules are tested by integrating Bessel functions and by solving a second-order elliptic eigenvalue problem with Dirichlet boundary conditions.

Goal-oriented a-posteriori estimation of model error as an aid to parameter estimation

In this work, a Bayesian model calibration framework is presented that utilizes goal-oriented a-posterior error estimates in quantities of interest (QoIs) for classes of high-fidelity models characterized by PDEs. It is shown that for a large class of computational models, it is possible to develop a computationally inexpensive procedure for calibrating parameters of high-fidelity models of physical events when the parameters of low-fidelity (surrogate) models are known with acceptable accuracy. The main ingredients in the proposed model calibration scheme are goal-oriented a-posteriori estimates of error in QoIs computed using a so-called lower fidelity model compared to those of an uncalibrated higher fidelity model. The estimates of error in QoIs are used to define likelihood functions in Bayesian inversion analysis. A standard Bayesian approach is employed to compute the posterior distribution of model parameters of high-fidelity models. As applications, parameters in a quasi-linear second-order elliptic boundary-value problem (BVP) are calibrated using a second-order linear elliptic BVP. In a second application, parameters of a tumor growth model involving nonlinear time-dependent PDEs are calibrated using a lower fidelity linear tumor growth model with known parameter values.

On Some Convergence Properties for Finite Element Approximations to the Inverse of Linear Elliptic Operators

This paper deals with convergence theorems of the Galerkin finite element approximation for the second-order elliptic boundary value problems. Under some quite general settings, we show not only the pointwise convergence but also prove that the norm of approximate operator converges to the corresponding norm for the inverse of a linear elliptic operator. Since the approximate norm estimates of linearized inverse operator play an essential role in the numerical verification method of solutions for non-linear elliptic problems, our result is also important in terms of guaranteeing its validity. Furthermore, the present method can also be applied to more general elliptic problems, e.g., biharmonic problems and so on.