Element Interiors Research Articles

Error analysis of implicit-explicit weak Galerkin finite element method for time-dependent nonlinear convection-diffusion problem

This paper focuses on the exploration of an implicit-explicit (IMEX) weak Galerkin finite element method (WG-FEM) applied to a one-dimensional nonlinear convection-diffusion equation. Based on a special weak form featuring two built-in parameters, we propose the fully implicit-explicit discrete WG finite element scheme. The diffusion term is treated implicitly, while the nonlinear convection term is treated explicitly. The WG-FEM utilizes locally piecewise polynomials of degree k to approximate the primal variable within the element interiors, along with piecewise polynomials of degree k+1 for the weak derivatives. Optimal error estimates in the L2 norm for the fully discrete scheme are derived in the theoretical analysis. Furthermore, we conduct numerical experiments to illustrate the effectiveness and accuracy of the proposed scheme.

A method for nonlinear electric-thermal coupling calculations of bushings based on unbiased gradient-free smooth domains.

High-voltage dry-type bushings, serving as the crucial junctions in DC power transmission, represent equipment with the highest failure rate on the DC primary side, underscoring the critical importance of monitoring their condition. Presently, numerical simulation methods are commonly employed to assess the internal state of bushings. However, due to limitations in the efficiency of multi-physics field computations, the guidance provided by numerical simulation results in the field of power equipment condition assessment is relatively weak. This paper focuses on solving the electrical-thermal coupling in high-voltage dry-type bushings. Addressing the most widely used tetrahedral mesh in numerical computations, we propose an efficient solution method based on the concept of "smooth domains." This method involves partitioning the volume centroids of the elements into multiple smooth domains within the computational domain. Electric and thermal conduction matrix calculations occur within these smooth domains, rather than within the grid or element interiors. This approach eliminates the need for traditional element mapping and complex volume integration. To demonstrate the effectiveness of this method, we use high-voltage dry-type bushings as a case study, comparing the performance of our approach with traditional finite element algorithms. We verify the algorithm's computational efficiency and apply it to the analysis of typical temperature anomalies in bushings, further illustrating its suitability for electrical equipment condition assessment.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

Full weak Galerkin finite element discretizations for poroelasticity problems in the primal formulation

This paper develops novel finite element solvers for linear poroelasticity problems on quadrilateral meshes. These solvers are based on the primal formulations of linear elasticity and Darcy flow. Specifically, the fluid pressure and solid displacement are approximated by scalar- or vector-valued polynomials of degree k≥0 separately in element interiors and on edges. The discrete weak gradients of these shape functions are established in the broken (vector- or matrix-version) Arbogast–Correa spaces for approximations of the classical gradients in the variational forms. These weak Galerkin spatial discretizations are combined with the implicit Euler or Crank–Nicolson temporal discretizations to develop locking-free numerical solvers that have optimal order (k+1) convergence rates in pressure, velocity, displacement, stress, and dilation. Rigorous analysis is presented and illustrated by numerical experiments on popular test cases.

Robust globally divergence-free Weak Galerkin finite element method for incompressible Magnetohydrodynamics flow

This paper develops a weak Galerkin (WG) finite element method of arbitrary order for the steady incompressible Magnetohydrodynamics equations. The WG scheme uses piecewise polynomials of degrees k(k≥1),k,k−1 and k−1 respectively for the approximations of the velocity, the magnetic field, the pressure, and the magnetic pseudo-pressure in the interior of elements, and uses piecewise polynomials of degree k for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. We give existence and uniqueness results for the discrete scheme and derive optimal a priori error estimates. We also present a convergent linearized iterative algorithm. Numerical experiments are provided to verify the obtained theoretical results.

A weak Galerkin finite element method for nonlinear convection-diffusion equation

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k+1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H1 and standard L2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings.

A weak Galerkin finite element method for 1D semiconductor device simulation models

In this paper, we study a weak Galerkin (WG) finite element method for semiconductor device simulations. We consider the one-dimensional drift–diffusion (DD) and high-field (HF) models, which involves not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. The main difficulties in the analysis include the treatment of the nonlinearity and coupling of the models. The weak Galerkin finite element method adopts piecewise polynomials of degree k for the approximations of electron concentration and electric potential in the interior of elements, and piecewise polynomials of degree k+1 for the discrete weak derivative space. The optimal order error estimates in a discrete H1 norm and the standard L2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems.

An Interface/Boundary-Unfitted EXtended HDG Method for Linear Elasticity Problems

An interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method of arbitrary order is proposed for linear elasticity interface problems on unfitted meshes with respect to the interface and domain boundary. The method uses piecewise polynomials of degrees \(k\ (\ge 1)\) and \(k-1\) respectively for the displacement and stress approximations in the interior of elements inside the subdomains separated by the interface, and piecewise polynomials of degree k for the numerical traces of the displacement on the inter-element boundaries inside the subdomains and on the interface/boundary of the domain. Optimal error estimates in \(L^2\)-norm for the stress and displacement are derived. Finally, numerical experiments confirm the theoretical results and show that the method also applies to the case of crack-tip domain.

Development of a muon tomography application with Micromegas detectors

The application of muon tomography method in small scale, for the imaging of a five-centimeter side lead cube, is being examined in the present work. The representation of the object is being achieved by using the transmission muography technique, in which the “free-sky” muon flux is compared to the muon flux measured within the acceptance of the detector. This comparison yields information about the absorption of the muons as they pass through the object under investigation. Also, a projective reconstruction method called “Back Projection” is tested by being applied on the data, providing information about the location of the object and its dimensions. This project has been carried out within the frame of EKATY programme, which aims to the innovative imaging of the subsurface of archaeological sites and the interior of structural elements of monuments in three and four dimensions.

Pressure robust hybrid mixed finite element method for Darcy–Forchheimer model

In this paper, we propose a pressure robust hybrid mixed finite element method for Darcy–Forchheimer model. The method uses piecewise polynomials of degrees for the velocity and k−1 for pressure in the interior of elements inside the domain, and piecewise polynomials of degree k for the numerical traces of the pressure on the inter-element boundaries. Our method can get the globally divergence-free approximation solution of velocity. Optimal error estimates are derived and the error estimates of velocity are pressure robust. Finally, numerical experiments confirm the theoretical results.

Feasibility study of muon tomography application in a non-invasive representation of tumulus

Within the frame of the EKATϒ programme, whose purpose is the innovative imaging of the subsurface of archaeological sites and the interior of structural elements of monuments in “three” and “four” dimensions, the applicability of Muon Tomography technique in the representation of a tumulus is tested in the present work. The scanning of its internal structure is accomplished by measuring the flux deficit of cosmic muon tracks in the presence of an object inside the tumulus, compared to the muon flux when traversing a uniform tumulus (transmission muography). The feasibility study of the method is achieved with a simulation of the tumulus geometry and the structure under investigation. Following the simulation process, a tracking telescope, consisting of four MicroMegas detectors and two trigger plastic scintillators, will be placed near Apollonia’s tumulus to collect data. For the specific latitude where the Apollonia’s tumulus is located, the energy and angular muon distribution at sea level is studied. Implementing the dimensions of the telescope in the simulation, the back-projection method is examined for the localization of the hidden object and the estimation of its dimensions. The method is tested for the telescope optimal position, placed under the tumulus, and the realistic one, placed near the tumulus at the level of its base.

A global residual‐based stabilization for equal‐order finite element approximations of incompressible flows

AbstractDue to simplicity in implementation and data structure, elements with equal‐order interpolation of velocity and pressure are very popular in finite‐element‐based flow simulations. Although such pairs are inf‐sup unstable, various stabilization techniques exist to circumvent that and yield accurate approximations. The most popular one is the pressure‐stabilized Petrov–Galerkin (PSPG) method, which consists of relaxing the incompressibility constraint with a weighted residual of the momentum equation. Yet, PSPG can perform poorly for low‐order elements in diffusion‐dominated flows, since first‐order polynomial spaces are unable to approximate the second‐order derivatives required for evaluating the viscous part of the stabilization term. Alternative techniques normally require additional projections or unconventional data structures. In this context, we present a novel technique that rewrites the second‐order viscous term as a first‐order boundary term, thereby allowing the complete computation of the residual even for lowest‐order elements. Our method has a similar structure to standard residual‐based formulations, but the stabilization term is computed globally instead of only in element interiors. This results in a scheme that does not relax incompressibility, thereby leading to improved approximations. The new method is simple to implement and accurate for a wide range of stabilization parameters, which is confirmed by various numerical examples.

Equivalence between the DPG method and the exponential integrators for linear parabolic problems

The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well established numerical methods for solving Partial Differential Equations (PDEs) and stiff systems of Ordinary Differential Equations (ODEs), respectively. In this work, we apply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general first order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D + time linear parabolic PDEs after discretizing in space by the finite element method.

A DPG-based time-marching scheme for linear hyperbolic problems

The Discontinuous Petrov–Galerkin (DPG) method is a widely employed discretization method for Partial Differential Equations (PDEs). In a recent work, we applied the DPG method with optimal test functions for the time integration of transient parabolic PDEs. We showed that the resulting DPG-based time-marching scheme is equivalent to exponential integrators for the trace variables. In this work, we extend the aforementioned method to time-dependent hyperbolic PDEs. For that, we reduce the second order system in time to first order and we calculate the optimal testing analytically. We also relate our method with exponential integrators of Gautschi-type. Finally, we validate our method for 1D/2D + time linear wave equation after semidiscretization in space with a standard Bubnov–Galerkin method. The presented DPG-based time integrator provides expressions for the solution in the element interiors in addition to those on the traces. This allows to design different error estimators to perform adaptivity.

Modeling of steep layers in singularly perturbed diffusion–reaction equation via flexible fine-scale basis

We present a new stabilized method for the diffusion–reaction equation which develops sharp boundary and/or internal layers for the reaction-dominated case (i.e. singularly perturbed case). The method relies on an improved expression for the stabilization parameter that is derived via the fine-scale variational formulation facilitated by the variational multiscale (VMS) framework. The proposed fine-scale basis consists of enrichment functions which may be nonzero at element edges. The derived stabilization parameter enjoys spatial variation over element interiors and along inter-element boundaries that helps model the rapid variation of the residual of the Euler–Lagrange equations over the domain. This feature also facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects. The method is able to better satisfy the maximum principle as compared to other existing methods. We present a priori mathematical analysis of the stability and convergence of the method for the diffusion–reaction equation. Optimal convergence on meshes comprised of linear triangles and bilinear quadrilateral elements are presented for smooth problems, as well as for problems with steep boundary layers. Stability and accuracy features of the method for problems with discontinuous forcing function, internal layers, and boundary layers are shown and its performance on unstructured and distorted meshes comprised of quadrilateral and triangular elements is highlighted.

Coupling Arbogast–Correa and Bernardi–Raugel elements to resolve coupled Stokes–Darcy flow problems

This paper presents a finite element method for solving coupled Stokes–Darcy flow problems by combining the classical Bernardi–Raugel finite elements and the recently developed Arbogast–Correa (AC) spaces on quadrilateral meshes. The novel weak Galerkin methodology is employed for discretization of the Darcy equation. Specifically, piecewise constant approximants separately defined in element interiors and on edges are utilized to approximate the Darcy pressure. The discrete weak gradients of these shape functions and the numerical Darcy velocity are established in the lowest order AC space. The Bernardi–Raugel elements (BR1,Q0) are used to discretize the Stokes equations. These two types of discretizations are combined at an interface, where kinematic, normal stress, and the Beavers–Joseph–Saffman (BJS) conditions are applied. Rigorous error analysis along with numerical experiments demonstrate that the method is stable and has optimal-order accuracy.

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

AbstractIn this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P0, P0, RT0). The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.

EXtended HDG Methods for Second Order Elliptic Interface Problems

In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise $C^2$ smooth interface. It uses piecewise polynomials of degrees $k$ $(k>= 1)$ and $k-1$ respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree $ k$ for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree $ k-1$ for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in $L^2$ norm and for the potential approximation in piecewise $H^1$ seminorm without requiring "sufficiently large" stabilization parameters in the schemes. In addition, error estimation for the potential approximation in $L^2$ norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results.

Element boundary terms in reduced order models for flow problems: Domain decomposition and adaptive coarse mesh hyper-reduction

In this paper we present a finite-element based reduced order model and, in particular, we consider two aspects related to the introduction of inter-element boundary terms in the formulation. The first is a domain decomposition strategy in which the transmission conditions involve boundary terms to account for non-matching meshes and discontinuous physical properties. The second is a coarse mesh hyper-reduction for which we propose an adaptive refinement driven by an a posteriori error estimator that contains element boundary terms. As the finite element full order model, the reduced order model is based on the Variational Multi-Scale framework, with sub-grid scales defined not only in the element interiors, but also on the inter-element boundaries. We present some examples of application using the incompressible Navier–Stokes equations and the Boussinesq approximation.

Penalty-Free Any-Order Weak Galerkin FEMs for Elliptic Problems on Quadrilateral Meshes

This paper presents a family of weak Galerkin finite element methods for elliptic boundary value problems on convex quadrilateral meshes. These new methods use degree $$k \ge 0$$ polynomials separately in element interiors and on edges for approximating the primal variable. The discrete weak gradients of these shape functions are established in the local Arbogast–Correa $$AC_k $$ spaces. These discrete weak gradients are then used to approximate the classical gradient in the variational formulation. These new methods do not use any nonphysical penalty factor but produce optimal-order approximation to the primal variable, flux, normal flux, and divergence of flux. Moreover, these new solvers are locally conservative and offer continuous normal fluxes. Numerical experiments are presented to demonstrate the accuracy of this family of new methods.