In this article, we use the stabilizer-free weak Galerkin (SFWG) finite element method on general polytopal meshes to solve the biharmonic equation. By decreasing the degree of the polynomials on the boundary of weak functions and modifying the definition of the weak Laplacian, this paper not only removes the stabilizer in the weak Galerkin (WG) numerical scheme, but also minimizes the number of unknowns in the scheme. The optimal orders of error estimates in the $H^2$ and $L^2$ norms are obtained and the convergence results are verified by some numerical examples.
In the present work, we propose two kinds of sixth-order numerical scheme for the Swift-Hohenberg (SH) equation. We adopt the implicit-explicit sixth-order scheme (BDF6) combined with the generalized scalar auxiliary variable (GSAV) method for temporal discretization and the Fourier spectral approach for spatial discretization. We rigorously prove the energy dissipation law of our proposed scheme and the $H^2$ norm error analysis. Furthermore, we promote the corresponding numerical scheme with the exponential scalar auxiliary variable (ESAV) method, which satisfies energy dissipation law. Finally, we show the substantial examples that verify the energy dissipation property and convergence of the two schemes.
In this study, we propose and analyze a first-order positivity-preserving numerical scheme, referred to as the jump-adapted drift-diffusion double implicit Milstein method, for solving a class of nonlinear jump-diffusion problems. The proposed method offers a more simple structure than the classic Milstein method for jump-diffusion problems, making it easier to implement on computers. By addressing the key challenges in strong convergence analysis stemming from non-Lipschitz continuity, weaker temporal regularity, and stochastic time partitioning, we rigorously prove that the proposed method achieves strong convergence with an optimal mean-square rate of order one. Moreover, under certain reasonable assumptions, we further demonstrate the positivity preservation of the proposed numerical method. Finally, numerical experiments are carried out to substantiate the validity of our theoretical results.
Voltage and current wave propagation on transmission lines is commonly described by the telegrapher's equations. However, in practical applications, the parameters of the telegrapher's equations - such as inductance and capacitance - often exhibit uncertainty due to manufacturing tolerances, material inhomogeneities, and environmental variations. This study presents a numerical framework for solving the telegrapher's equations with uncertain parameters using a combination of Polynomial Chaos Expansion and the Finite Difference Time Domain (FDTD) method. The proposed approach enables efficient uncertainty quantification while maintaining computational tractability. In addition to the numerical formulation, the conditional stability of our method is analyzed, and the discrete dispersion relation is derived and compared with the continuous dispersion relation. The results demonstrate that the PC-FDTD method is a robust and accurate tool for modeling wave behavior in transmission lines under parameter uncertainty.
Resolving multiscale equations in heterogeneous materials presents significant computational challenges arising from rapid spatial oscillations and random variations in solutions induced by intricate microstructural configurations. This study proposes and analyzes a two-stage stochastic homogenization framework designed to efficiently compute homogenized solutions for multiscale diffusion equations. The methodology unfolds through two distinct phases. In the first stage, each realization of the random microstructure undergoes spatial homogenization through a representative volume element (RVE)-based approach, effectively replacing the original multiscale diffusion equation with a random counterpart featuring piecewise constant coefficients. Through ensemble-based Monte Carlo (EMC) averaging of diffusion coefficients, we reformulate the random diffusion equation into a deterministic diffusion problem with a random source term in the second stage. This critical reformulation enables the implementation of an efficient fixed-point iteration scheme for solving the resultant constant-coefficient diffusion equation. In addition, the convergence of the homogenized solution to the solution of multiscale diffusion equation is proved. Numerical examples are provided to demonstrate the ability and accuracy of the proposed method.
Two combined methods for computing solutions of time-varying semilinear differentialalgebraic equations (descriptor systems) are obtained. When constructing the methods, timevarying spectral projectors which can be found numerically are used. This enables one to numerically solve the differential-algebraic equation (DAE) in the original form without additional analytical transformations. The convergence and correctness of the developed methods are proved. The methods are applicable to the semilinear DAEs with the continuous nonlinear part which may not be differentiable in time. The global Lipschitz condition and other conditions of this kind are not used in the presented theorems on the global solvability of DAEs and on the convergence of the methods. This extends the scope of the methods. The obtained theorems ensure both the existence of a unique global exact solution and the convergence of the methods, which enables one to compute an approximate solution on any given time interval. Numerical examples illustrating the capabilities of the methods and their effectiveness in various situations are provided. To demonstrate the practical application of the obtained methods and theorems, the numerical and theoretical analyses of mathematical models of the dynamics of electric circuits are carried out. It is shown that their results are consistent.
The Q1-finite element spaces, in any finite d-dimension, are equipped with the discrete L'2h-inner product generated by the simple row-sum mass lumping. The equivalence of the discrete L'2h-norm and the L2-norm on the Q1-finite element spaces is uniform in mesh size h, in both cases of uniform and nonuniform partitions. Several representation formulae for these norms are derived. Using these, accurate bounds between these two norms are obtained, which is our major contribution. Examples show that these bounds are sharp. As an important application, the equivalence is established between discrete h1h-norm and H1-norm. Numerical results are presented.
This paper presents a boundary corrections mixed finite element method for second order elliptic equations with the non-homogeneous Neumann boundary condition on curved domains. A key feature of the boundary value corrections is the shift from the true boundary to a surrogate boundary, which avoids numerical integration formula on curved elements. We consider the high-order Raviart-Thomas element (RTk) of degree k >= 1 on triangular meshes, achieving an O(hk+1/2) convergence in the L2-norm estimate for the velocity field and an O(hk) convergence in the H1-norm estimate for the pressure. Finally, numerical experiments validate our theoretical results.
Electroporoelasticity equations comprise of Maxwell's equations and Biot's equations, playing an important role in geophysical areas such as oil-gas exploration and earthquake early warning. The development of electromagnetic waves and elastic waves presents distinct time scales due to the multi-physics nature. In this paper, we propose a multi-time stepping numerical algorithm to approximate electroporoelasticity equations, in which we use a smaller time step to compute Maxwell's equations and a larger time step to calculate Biot's equations. We prove the stability of this algorithm and derive its error estimates. Numerical experiments are conducted to demonstrate the theoretical analysis.
In this paper, we study an ultra-weak discontinuous Galerkin (UWDG) method for spatial discretization to solve the drift-diffusion (DD) model of one-dimensional semiconductor devices. Optimal error estimates are obtained using a special projection for both the semi-discrete and fully discrete UWDG schemes with smooth solutions. In the fully discrete UWDG scheme, we use an explicit third-order total variation diminishing Runge-Kutta method, which ensures stability under general temporal-spatial conditions. Numerical simulations are also performed to verify the analysis.
Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs). However, their effectiveness deteriorates when applied to singularly perturbed problems, where solutions exhibit steep gradients and boundary layers confined to narrow regions features that standard PINNs architectures often fail to capture. To overcome these obstacles, we propose a novel, mesh-free deep neural networks (DNNs) method that incorporates asymptotic information through a systematic decomposition of the solution into smooth and sharp components. By capitalizing on the ability of DNNs to excel at approximating smooth functions, our approach constructs a series of moderately sized networks tailored to different elements of the solution. This strategy enables uniform approximation accuracy across a wide range of perturbation parameters, maintaining robustness and efficiency even in the extreme regime where the perturbation parameter is as small as 10-16, near machine precision. Key advantages of the proposed method include its conceptual simplicity, full independence from mesh requirements, and ease of implementation. Extensive numerical experiments confirm that our approach delivers significantly improved accuracy and efficiency compared to standard PINNs, demonstrating its potential as a robust and versatile framework for tackling singularly perturbed problems.
In the first part of this paper, the uniqueness of a strong solution is established for the Vlasov-unsteady Stokes problem in 3D. The second part deals with a semi discrete scheme, which is derived as a result of spatial discretization of the coupled system of Vlasov and Stokes equations for the 2D problem by discontinuous Galerkin methods, while keeping temporal variable continuous. The proposed semi-discrete scheme preserves both mass and momentum conservation properties. Based on the orthogonal L-2 as well as the Stokes projections, error estimates in the case of smooth compactly supported initial data are derived by employing a variant of nonlinear Gronwall's lemma in a crucial way. Moreover, the generalization of error estimates to 3D problem is also briefly discussed. Finally, using a time splitting algorithm as the phase space is four dimensional, some numerical experiments are conducted, whose results confirm our theoretical findings.
This manuscript proposes, analyzes, and illustrates an iso-parametric finite element splitting extrapolation method for accurately and efficiently solving the second-order semi-linear and quasi-linear parabolic equations with curved boundaries. The design of multiple grid size parameters is based on an appropriate domain decomposition of the original problem domain and iso-parametric mapping, hence provides more flexibility to form the grid and constructs the extrapolation schemes. To reach the same level of accuracy of a globally refined grid (or even better accuracy) with less expenses, we only need to solve a group of smaller discrete problems on a set of locally refined grids, instead of solving a much larger discrete problem on the globally refined grid. To develop such an accurate and efficient scheme, multi-parameter expansions for the semi-discrete and fully discrete iso-parametric finite element errors are first proved. Then the extrapolation idea can be utilized to construct the splitting extrapolation schemes based on the designed multiple grid size parameters. A posterior error estimates are provided for the splitting extrapolation solutions. Numerical examples are also provided to illustrate the obvious accuracy improvement from the splitting extrapolation schemes.
We provide a unified analysis framework for discretized Volterra integrodifferential equations by considering the 9-type convolution quadrature, where different 9 corresponds to different schemes. We first derive the long-time lCO stability of discrete solutions, and then prove a discrete Wiener-Levy theorem to support the analysis of long-time l1 stability. The methods we adopt include the integral transforms in the Stieltjes sense, the complex analysis techniques, and a linear algebra approach for an indirect estimate of intricate terms. Meanwhile, we relax the commonly-used regularity assumption of the initial data in the literature by novel treatments. Numerical simulations are performed to substantiate the theoretical findings.
This study explores the application of Peridynamic (PD) theory in ground fissure modeling and simulation, focusing on the Songzhuang region in Beijing. Traditional methods often encounter difficulties in accurately capturing fissure formation and propagation due to discontinuities at the tips of the crack. The PD model, formulated through integral equations, effectively addresses these issues. It ensures model continuity and considers interactions within a non-local neighborhood of points. First, the study validates the numerical algorithm of the PD model through a two-dimensional tensile test of a plate with a central crack and a simulation of hidden ground fissures in Xi'an. The PD model is then applied to the Songzhuang area. The study integrated field drilling data to simulate fissure development. The results show fissures that grow in the expected direction, with significant subsidence and shear deformation on both sides. These findings provide new information on the impacts of ground fissures. Despite challenges such as high computational demands and the need for further model refinement, the study confirms the broad applicability of PD theory to geohazard mitigation. It also demonstrates the potential of the theory in improving urban planning and improving disaster response strategies.
In this work, we will study, from both analytical and numerical points of view, a nonlo cal problem involving a thermovisco elastic beam which has been modeled by using the type II thermal law. In the first part, we will show that this problem has a unique solution and that the solutions decay exponentially by using the theory of linear semigroups. We will also prove that the semigroup of contractions is not differentiable and the impossibility of localization, that is, we will obtain that the unique solution which can vanish in an open nonempty set is the null solution. In the second part, we will focus on the numerical approximation of a variational formulation of the thermomechanical problem. By using the finite element method and the implicit Euler scheme to approximate the spatial variable and to discretrize the time derivatives, respectively, a fully discrete scheme will be introduced. Then, we will prove a discrete stability property and we will provide an a priori error analysis. The linear convergence of the approximations will be deduced whenever the continuous solution is regular enough. Finally, some numerical results will be presented to demonstrate the numerical convergence and the exponential decay of the discrete energy.
In this paper, we study the blood solute dynamics model to understand the relationship between the widespread pathologies of the vascular system, the specific features of the blood flow in a diseased district, and the effect of the flow pattern on the transfer processes of solute within arterial lumen and wall. The proposed finite element algorithm is based on the second-order backward differentiation formula and the explicit treatment of the coupling terms, which allow us to solve the decoupled Navier-Stokes equations, advection-diffusion equation, and pure diffusion equation at each time step. We derive the unconditional and long-time stability in the sense that the solution remains uniformly bounded in time, leading to uniform time error estimation. The long-time accurate behavior is one of the most desirable physical processes for the development of cardiovascular diseases that occurs over long-time scale. To validate the proposed method and demonstrate the exclusive features of the blood solute dynamical model, we perform four numerical experiments. Moreover, the impact of the development of atherosclerosis lesion and abdominal aortic aneurysm are studied by illustrating the complicated flow characteristics, streamlines, pressure contours, solute concentration, wall shear stress, and long-time accuracy on the several geometrical setups for the physiological interests.
We introduce the Energy Dissipation Rate guided Adaptive Sampling (EDRAS) strategy, a novel method that substantially enhances the performance of Physics-Informed Neural Networks (PINNs) in solving thermodynamically consistent partial differential equations (PDEs) over arbitrary domains. EDRAS leverages the local energy dissipation rate density as a guiding metric to identify and adaptively re-sample critical collocation points from both the interior and boundary of the computational domain. This dynamical sampling approach improves the accuracy of residual-based PINNs by aligning the training process with the underlying physical structure of the system. In this study, we demonstrate the effectiveness of EDRAS using the Allen-Cahn phase field model in irregular geometries, achieving up to a sixfold reduction in the relative mean square error compared to traditional residual-based adaptive refinement (RAR) methods. Moreover, we compare EDRAS with other residual-based adaptive sampling approaches and show that EDRAS is not only computationally more efficient but also more likely to identify high-impact collocation points. Through numerical solutions of the Allen-Cahn equation with both static (Neumann) and dynamic boundary conditions in 2D disk- and ellipse-shaped domains solved using PINN coupled with EDRAS, we gain significant insights into how dynamic boundary conditions influence bulk phase evolution and thermodynamic behavior. The proposed approach offers an effective, physically informed enhancement to PINN frameworks for solving thermodynamically consistent models, making PINN a robust and versatile computational tool for investigating complex thermodynamic processes in arbitrary geometries.
In this article we develop a Physics Informed Neural Network (PINN) approach to simulate ice sheet dynamics governed by the Shallow Ice Approximation. This problem takes the form of a time-dependent parabolic obstacle problem. Prior work has used this approach to address the stationary obstacle problem and here we extend it to the time dependent problem. Through comprehensive 1D and 2D simulations, we validate the model's effectiveness in capturing complex free-boundary conditions. By merging traditional mathematical modeling with cutting-edge deep learning methods, this approach provides a scalable and robust solution for predicting temporal variations in ice thickness. To illustrate this approach in a real world setting, we simulate the dynamics of the Devon Ice Cap, incorporating aerogeophysical data from 2000 and 2018.
