We investigate the energy-based discontinuous Galerkin (EDG) methods for solving second-order wave equations. The standard EDG formulation produces spurious oscillations near solution discontinuities and yields incorrect wave speeds when the initial data contains a discontinuity. To address these issues, we introduce an oscillation-free approach, augmented with an additional penalty term, to develop the OF-EDG method. The new formulation effectively suppresses spurious oscillations near discontinuities while preserving high-order accuracy for smooth solutions. We establish stability analysis and provide a priori error estimates for several common numerical flux choices. Through a series of numerical experiments, we demonstrate optimal convergence for smooth solutions and confirm the robustness of the OF-EDG method in maintaining oscillation-free behavior for nonsmooth solutions, both for linear wave equations and those with nonlinear source terms. Furthermore, we highlight the importance of the penalty term for ensuring convergence to the true solution when the initial data contains discontinuities.

Simulation of fate and transformation of nitrogen compounds in groundwater using meshless weak strong (MWS) form method.

Unconditionally superconvergent error analysis of an energy-conservative Galerkin method for the nonlinear Schrödinger equation with wave operator

An unconditionally stable hybridizable-embedded discontinuous Galerkin method for the phase field crystal equation

Entropy stable reduced order modeling of nonlinear conservation laws using discontinuous Galerkin methods

An embedded-hybridized discontinuous Galerkin method for the electrohydrodynamics system

In this paper, we concentrate on the superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear time-dependent fourth-order equations. The numerical flux for the nonlinear convection term is chosen as the generalized local Lax–Friedrichs flux, and the generalized alternating fluxes are employed for the fourth- and second-order terms, which are beneficial for long time simulations with a slower error growth due to the adjustable numerical viscosities. For nonlinear fourth-order equations with periodic boundary conditions, by using generalized Gauss–Radau projections, a modified projection and correction functions, we show a superconvergent bound for the interpolation errors. Then, by designing the numerical initial condition as an interpolation function of the third-order derivative, we derive supercloseness and thus superconvergence results, no matter whether the wind direction is fixed or not. Specifically, for polynomials of degree k, we obtain (2k+1)th order superconvergence for the numerical flux and cell averages, (k+2)th order superconvergence at generalized Radau points, and (k+1)th order for the error derivative at generalized Radau points, followed by a supercloseness result of order k+2 between the generalized Gauss–Radau projections and the numerical solutions. The superconvergence results are extended to the case with mixed boundary conditions when the wind direction is fixed. A series of numerical examples, including various boundary conditions and nonlinear terms, together with long time simulations, are provided to validate the theoretical results and demonstrate the effectiveness of the method.

ABSTRACT An online adaptive reduced order model (ROM) of the discontinuous Galerkin (DG) method is developed for predicting unsteady scale‐resolved flow simulation. The least‐squares Petrov‐Galerkin (LSPG) projection is chosen as the baseline ROM framework, along with typical hyperreduction techniques for acceleration. Since LSPG requires multiplication operations of the Jacobian and the basis, a Jacobian‐free approach is proposed for forming the low‐dimensional ROM system, to keep consistent with the Jacobian‐free strategy of the original implicit DG method. Then, a comprehensive online adaptation algorithm of the LSPG model is developed by updating the basis and sampling elements with snapshots generated by the full‐order DG solver in an efficient way. The key idea for the adaptation is to update the ROM with the most recent flow information to predict the unseen features. Given a set of parameters, the proposed algorithm firstly runs the full‐order DG solver for a short period, secondly generates the initial basis and reduced mesh, and finally runs the adaptive ROM for future‐state predictions, which enables the model to possess predictive capability. Several benchmark cases have been conducted for verification and comparison to static ROMs. The chosen cases involve typical challenges, that is, transportation and discontinuity, for static ROMs, including the isentropic vortex convection, the Sod shock tube, the Kelvin‐Helmholtz instability, and the two‐dimensional Riemann problem. The results demonstrate that the adaptive ROM is able to effectively address the above challenges encountered by its static counterpart from a predictive perspective while achieving reasonable acceleration.

In adherent cells, actomyosin contractility is regulated mainly by the RhoA signaling pathway, which can be controlled by optogenetics. To model the mechanochemical coupling in such systems, we introduce a finite element framework based on the discontinuous Galerkin method, which allows us to treat cell doublets, chains of cells, and monolayers within the same conceptual framework. While the adherent cell layer is modeled as an actively contracting viscoelastic solid on an elastic foundation, different models are considered for the Rho pathway, starting with a simple linear chain that can be solved analytically and later including direct feedback that can be solved only numerically. Our model predicts signal propagation as a function of coupling strength and viscoelastic timescales and identifies the conditions for optimal cell responses and wave propagation. In general, it provides a systematic understanding of how biochemistry and mechanics simultaneously contribute to the communication of adherent cells.

This research presents an analytical investigation into the nonlinear vibrational behavior of graphene nanoplatelet-reinforced polymer (GPL-R) plates subjected to external excitation. The novelty of the proposed methodology lies in establishing a direct test-to-dynamics framework in which tensile-test-derived Mooney-Rivlin constants are embedded into the forced nonlinear vibration formulation of GPL-reinforced polymer plates, rather than treating the nanocomposite as an equivalent linear elastic or purely homogenized material. To develop a more realistic constitutive model for the nanocomposite plate, this study integrates experimentally derived hyperelastic parameters of GPL-epoxy nanocomposites into a nonlinear plate vibration model governed by a Mooney-Rivlin strain-energy formulation. Following the derivation of the nonlinear governing equations of motion via Hamilton's principle, Galerkin's method is applied to discretize the system. The effective mechanical properties of the nanocomposite are obtained through experimental tensile testing performed on specimens containing varying concentrations of graphene nanoplatelets. Subsequently, the discretized equations are solved numerically to characterize the nonlinear dynamic behavior of the system. To this end, the influence of key parameters on various nonlinear phenomena is assessed using time-history responses, phase-plane portraits, Poincaré maps, and frequency-response curves. The experimentally calibrated analytical-numerical methodology avoids arbitrary hyperelastic-parameter assumptions and enables direct transfer of the measured GPL-dependent nonlinear material behavior into the vibration model. The results demonstrate that increasing the GPL content up to 1.0 wt% increases the resonant frequency by 57.9% and reduces the maximum vibration amplitude by 50% compared with pure epoxy. Furthermore, the formulation captures a softening-to-hardening transition governed by the competition between Mooney-Rivlin material nonlinearity and von Kármán membrane stretching. Accounting for hyperelastic material behavior is shown to be crucial for accurately predicting the dynamic response, particularly at larger amplitudes, where linear elastic models overestimate deflections.

In this study, the energy expressions of a perforated plate with hysteresis-type elastic-dissipative characteristics subjected to kinematic excitations are determined, and based on them, the differential equation of motion is formulated using the second-order Lagrange equation. The dissipative properties of the plate material are described using the expressions derived from the Pisarenko-Boginich hypothesis, and are incorporated through coefficients in explicit form by means of the harmonic linearization method. The cut-out extracted from the rectangular plate is also assumed to be rectangular in shape, with its sides parallel to those of the plate, and its location considered arbitrary within the plate domain. The kinetic and potential energies are expressed separately for the plate and the corresponding cut-out region, and, based on the equality of displacements along the cut-out boundary, the necessary compatibility relations are established. As a result, both the kinetic and potential energies are ultimately expressed solely in terms of the plate deflection. The mode shapes of the perforated plate are assumed to be orthogonal, and by applying the Bubnov-Galerkin method, the governing differential equation of motion is reduced to a simplified form.

Influence of thermal tuning and graded thickness for sound transmission loss in functionally graded plates based on Galerkin method

p-multigrid method for the discontinuous Galerkin discretization of elliptic problems with discontinuous coefficients

A priori error estimates based on lyapunov for the FitzHugh-Nagumo model via interior penalty discontinuous Galerkin method

Hybridizable discontinuous galerkin methods for thermo-poroelastic systems

A modified weak Galerkin method for the biharmonic equation

This paper presents an enrichment method for the case of re-entrant corner singularities in linear elastic problems but with no enrichment of the finite element solution space. Instead an a posteriori error estimator, with gradient descent, is used to determine the approximate solution of the coefficients for the enrichment functions about each re-entrant corner. The method then approximately removes the singularities from the problem, increasing its regularity. As a result is that exponential convergence of the error can be achieved with uniform refinement in polynomial order. Almost no improvement in the error is expected or observed if uniform refinement in p is used. The approach is termed the Celatus method as the singularities are hidden from view . As problems are made regular it is shown that exponential convergence rates are observed when the Celatus method is combined with h p -adaptivity ( h p -Celatus), requiring far fewer degrees of freedom compared, by orders of magnitude, to traditional finite element analysis with h p -adaptivity. Furthermore each term for the enrichment functions for every re-entrant corner can be evaluated independently. Therefore the method can be implemented in an inherently parallel way. The proposed approach offers an ≈ 10 times reduction in computation time for the same accuracy compared to standard finite element analysis with h p -adaptivity. Additionally, since the solution space is not enriched and is always polynomial, the issue of having near singular matrices does not exist for the Celatus method. The discontinuous Galerkin finite element method is used here, but all equations and methodology are equally applicable to the continuous Galerkin method. • Enrichment of solution achieved without enriching the finite element solution space. • Each corner’s enrichment terms are evaluated independently for parallel execution. • Requires orders of magnitude fewer degrees of freedom than traditional hp-adaptive FEM. • Singularities are removed, converting the problem from non-smooth to smooth, providing convergent results with increasing polynomial order only. • Method avoids issues with near-singular matrices common in enriched methods.

An adaptive node removal strategy for faster convergence using evolutionary structural optimization and Element Free Galerkin method

This paper investigates a fourth-order parabolic equation with double logarithm nonlinearities. Using the Galerkin method, we establish the global existence and uniqueness of weak solutions. By classifying the initial energy, we derive blow-up criteria of solutions and obtain explicit upper bounds for the blow-up time of solutions. Moreover, we present new threshold criteria for the extinction and non-extinction behavior of solutions. Specifically, we give the upper bound and lower bound for the extinction rate of solutions. These results generalize and improve some earlier related results in the literature.

In this article, we investigate least squares-based weak Galerkin methods for numerically approximating the solution of the Maxwell interface problems having non-homogeneous jumps across the interface, posed in Lipschitz continuous domains in 2D/3D with a C 2 interface. Rigorous convergence analysis has been carried out to obtain super-convergence of the errors in a discrete energy norm. Some numerical experiments are provided to support the theoretical conclusions.