This study develops a residual-to-error analysis for second-order Singularly Perturbed Volterra Integro-Differential Equations (SPVIDEs) with boundary conditions. Building on the established auxiliary-output PINN principle, the Volterra integral is represented by an auxiliary network output, and the primary and auxiliary approximations are constrained through two coupled residuals. The novelty of the present work does not lie in this auxiliary-output construction itself, but in the analytical chain linking the auxiliary residual to Volterra-defect control and, subsequently, to a conditional ℒ^∞ solution-error estimate. More precisely, the Volterra defect is defined as the discrepancy between the learned auxiliary output and the Volterra operator applied to the learned primary solution. We show that this defect is directly controlled by the auxiliary residual. Combining the resulting defect estimate with the primary-equation residual and a generalized Grönwall–Bellman argument yields a problem-specific conditional ℒ^∞ bound whose constants explicitly reflect the effects of the perturbation parameter, the coupling coefficient, and the kernel bound. For numerical assessment, the boundary conditions and the auxiliary initial condition are imposed exactly through hard-constrained trial functions. A dual-output PINN is trained using Adam followed by L-BFGS, and its performance is evaluated through discrete maximum errors and worst-case errors over the tested perturbation set. The results show high accuracy for smooth and moderately perturbed problems. In contrast, the present global PINN is not robust in the strongest boundary-layer regimes, where the maximum errors remain comparatively large and do not decrease monotonically under collocation refinement.
We study the time-dependent Navier–Stokes system coupled with the heat equation under nonlinear Tresca boundary conditions in two dimension. This frictional slip boundary condition is characterized by a subdifferential property. Employing an effective regularization technique, we reformulate the continuous variational inequality into an equivalent equality problem. We establish the existence and and conditional uniqueness of weak solutions. Furthermore, we develop a fully discrete scheme based on the characteristic method and provide optimal error estimates for velocity, pressure, and temperature. We end the paper with numerical simulation validating the theoretical results.
We present a monolithic divergence-conforming virtual element method for the linear fluid–structure interaction (FSI) problem with a thick structure. This problem consists of the time-dependent Stokes equations in the fluid domain coupled with the linear elastodynamics equations in the solid region. To keep the divergence be conforming, the global velocity is discretized by the H(div) virtual element with an additional polynomial space defined on the element edges to approximate the tangential trace of the velocity, while the pressure is discretized by the discontinuous piecewise polynomials. Such spatial approximation combined with the Crank-Nicolson technique for the temporal discretization leads to a fully discrete scheme that produces an exactly divergence-free fluid velocity approximation. The a priori error estimates of the fully discrete scheme are well established, where the error bound is robust to the physical parameters. Finally, numerical examples are provided to verify that our method is not affected by the added mass effect and is robust to physical parameters. MSC codes are 65N30, 65N12.
This paper presents a novel computational framework for solving the Rosenau–Burgers–biharmonic equation, a high-order nonlinear partial differential equation that models wave propagation in dispersive media with dissipative effects. The proposed approach combines Isogeometric Analysis (IGA) with high-order Explicit first stage, Singly Diagonally Implicit Runge–Kutta (ESDIRK) temporal discretization to address the significant challenges posed by the fourth-order biharmonic term. The IGA formulation leverages the inherent high-order continuity of Non-Uniform Rational B-Splines (NURBS) to achieve H^2 -conformity without requiring mixed formulations or complex C^1 -continuous finite elements. The adaptive refinement is realised through hierarchical B-splines (HB-splines), enabling local resolution of steep fronts while preserving the required high-order continuity. We establish the energy stability of the semi-discrete scheme and demonstrate optimal convergence rates in both space and time through rigorous numerical experiments. The method’s performance is validated through comparisons with traditional finite element approaches, adaptive refinement studies, and long-time dynamic simulations that confirm excellent agreement with asymptotic theoretical predictions. Our results show that the integrated IGA-ESDIRK framework offers superior accuracy, computational efficiency, and structure-preserving properties compared to existing numerical methods for this class of problems.
This paper investigates the application of the Parareal framework to a class of linear dissipative differential equations with a single constant time delay. A direct application of standard Parareal methods to delay differential equations typically results in nonuniform communication patterns due to the exchange of historical data. To overcome this difficulty, we propose a task allocation strategy based on delay intervals, with coarse and fine propagators defined using an implicit time discretization. The full time domain (0,Kτ ] is divided into KN coarse subintervals of size τ /N , with N subintervals in each delay interval of length τ . Fine propagations are performed in parallel using P=N processors, while the Parareal correction is applied sequentially across successive delay intervals. This strategy localizes the treatment of the delay term and avoids additional data exchange. From a theoretical perspective, we first establish a relationship between the coarse-grid and fine-grid solutions and then derive convergence estimates for the Parareal scheme. The resulting convergence behavior is influenced by the delay structure and stability properties of the propagators. We also define the speedup for an efficiency assessment. Numerical experiments are presented to illustrate the convergence behavior of the method for several examples.
In this paper we consider some computational applications of the conjugate Appell-Simpson polynomial sequences, recently introduced. Particularly, the Umbral-Simpson interpolation is studied both in the continuous and discrete case. Two new classes of interpolatory quadrature rules, which include the classic 1/3 Simpson formula and the midpoint formula, are obtained. A new operator of Jakimovski and Leviatan type is derived for the approximation of real functions on the semi-infinite axes. Several numerical examples are given. The results of this paper appear to be new.
The approximation of tensors in a low-para metric format is a crucial component in many mathematical modelling and data analysis tasks. Among the widely used low-parametric representations, the canonical polyadic (CP) decomposition is known to be very efficient. Nowadays, most algorithms for CP approximation aim to construct the approximation in the Frobenius norm; however, some applications require entrywise approximation. In this paper, we propose the alternating minimization method for constructing low-rank approximations of tensors in the CP format in the Chebyshev norm. Through an extensive evaluation, we demonstrate the effectiveness of the algorithm. We also analyze the proposed algorithm for rank-1 approximations and introduce the notion of d-way alternance. We show that the presence of a d-way alternance is a necessary condition for optimal approximation and that all limit points of the alternating minimization method satisfy this condition. Based on this analysis, we derive a modification of the algorithm that empirically shows promising results for constructing of the optimal rank-1 approximations.
Ill-posed inverse problems are pervasive across scientific and engineering disciplines, with solutions highly sensitive to data perturbations. Among regularization strategies to mitigate this instability, iterative methods have proven effective. In this paper, we propose an accelerated iterative regularization strategy for solving ill-posed linear operator equations, inspired by nonstationary iterated Tikhonov regularization. The method incorporates a priori and a posteriori parameter selection rules, both of which yield optimal-order error estimates. Compared to existing iterative approaches, the proposed strategy significantly reduces the number of iterations required for convergence under suitable stopping criteria. Numerical experiments validate its efficacy, demonstrating robust performance in solving ill-posed problems. Furthermore, the method’s adaptability is showcased through applications to image restoration, highlighting its practical relevance.
In this paper, we study a numerical method for solving the time-fractional Allen-Cahn equation. A fully discrete scheme is developed using an ultra-weak discontinuous Galerkin method for spatial discretization, and the temporal discretization is performed using the L1 method on graded meshes. A rigorous theoretical analysis is presented, establishing unique solvability, energy stability, and optimal a priori error estimates. The efficiency of the proposed scheme is demonstrated through numerical examples. Additionally, key physical properties such as the energy dissipation law, maximum principle, phase separation, and coalescence phenomena are investigated through simulations.
We propose a mixed formulation along with a regularization method to solve the Poisson equation under the Tresca friction law, and we provide a rigorous theoretical analysis, including the well-posedness and the error estimates on regularization parameter (and mesh size for discretization), in both continuous and discrete settings. We design two iteration algorithms to solve the discrete mixed regularization problem, and investigate the convergence. Several numerical experiments are conducted to verify the theoretical convergence rates and the effectiveness of the proposed algorithms.
We present a method for computing local compact finite difference (FD) formulations by combining polynomials and radial basis functions (RBFs). We explore two different methods for computing compact FD differentiation weights. The first method involves combining infinitely smooth Multiquadric (MQ) and Gaussian (GA) RBFs with polynomials, while the second method combines polyharmonic spline (PHS) with polynomials. Although we proposed the approach for a general node layouts, for specific stencils, we derive analytical weights for the first and second derivatives and use them to establish the truncation error equations. Our analysis shows that combining MQ or GA with polynomials yields compact FD formulas of the highest possible order on several important specific stencils. Ultimately, the computational outcomes underscore the efficacy and applicability of the presented methods.
Many partial differential equations can be solved by converting into a pseudo-differential operator equation. But when the data is noisy, the numerical computation of the solution is unstable. In this work, we consider a modified fractional Tikhonov regularization scheme along with parameter choice rules for solving pseudo-differential operator equations. A better rate of convergence for this scheme is demonstrated along with the parameter choice rules. Several examples as well as numerical results are also discussed to illustrate the viability of the scheme.
Large and little Schröder matrices as well as with their inverses are triangular matrices arising in Combinatorics. Through their total positivity and bidiagonal decomposition it is shown that their singular values, inverses and some associated linear systems can be solved with high relative accuracy. Numerical experiments confirm the theoretical results.
Recently, Benzi et al. proposed several iterative methods for solving the double saddle point system arising from the modeling of liquid crystal directors using finite elements. To evaluate the structured backward stability of these numerical algorithms, in this paper we consider the structured backward error analysis for the double saddle point system. Based on the structured backward error, we also present a new termination criterion used in the iterative method to verify whether the current iterative solution is an exact solution of the acceptable perturbed structure-preserving double saddle point system. Numerical examples show that the structured backward error can be easily used to test the stability of practical algorithms and the new stopping criterion is more suitable for the iteration methods for solving the double saddle point problems than the typical ones.
We develop and analyze a robust Arrow–Hurwicz (AH) iterative method for the numerical solution of steady Boussinesq flows with nonhomogeneous partitioned Dirichlet boundary conditions. Although a direct AH formulation may be applied to the momentum equation alone, we demonstrate that incorporating an AH-type update for the temperature equation is crucial for stability and convergence in buoyancy-driven systems, particularly at high Rayleigh numbers. The resulting Improved Arrow–Hurwicz (IAH) scheme avoids solving saddle-point systems at each iteration and yields a fully decoupled algorithm with low computational cost per step. We establish existence, uniqueness, uniform boundedness, and convergence under standard small-data assumptions, and provide corresponding error estimates for the finite element discretization. Extensive two- and three-dimensional numerical experiments verify the theoretical findings, demonstrate significant acceleration over the alternative AH scheme and the Penalty–Picard iteration, and confirm robust convergence in high–Rayleigh number regimes. The proposed method offers a scalable and efficient solver for steady natural convection and provides a promising alternative to continuation-based approaches traditionally used for high–Rayleigh flows.
In this paper, a numerical scheme is proposed and analyzed for Liu-Wu model: Cahn–Hilliard equation with Cahn-Hilliard type dynamic boundary conditions (Liu and Wu, Arch Ration Mech Anal 233(1):167–247, 2019). Inspired by the convex-concave decomposition of the Cahn–Hilliard free energy, prescribed with Neumann boundary condition, we propose a regularized second-order temporal discretization for the coupled PDE system. The unique solvability, unconditional dissipation of a modified free energy and second order convergence analysis (in the H^-1 norm) are theoretically established. Finally, several numerical experiments are presented to illustrate the effectiveness of the proposed numerical scheme.
The time-fractional Oseen equations are studied in this article, where the time-fractional derivative is in the sense of Caputo of order α∈ (0,1) . The L1 formula with uniform and nonuniform meshes is applied to approximating the temporal fractional derivative, and the local discontinuous Galerkin method is developed to approach the space derivative. The numerical stability and the error estimates of the constructed schemes concerning velocity, stress (gradient of velocity), and pressure are shown. Moreover, the fast L1 algorithm is developed to accelerate the evaluation of the Caputo derivative. Finally, numerical examples are presented to testify the effectiveness of the derived algorithms. Mathematics Subject Classification. 65M60, 35R11, 65M12.
In this paper, we deal with the stability of equilibria arising from both semi-discrete and fully discrete approximations of a Gurtin-MacCamy model with infinite age span. Numerical solutions obtained through both the discontinuous Galerkin semidiscretization with piecewise constant approximation in age and the partitioned implicit-explicit Euler method are shown to replicate the stability of the equilibria. For the discontinuous Galerkin semi-discrete processes, a numerical reproduction number is introduced to derive conditions that ensure the existence of a numerical nontrivial equilibrium. Meanwhile, by linearizing the nonlinear model, the local stability of numerical equilibrium distributions is discussed. As an application to the Gurtin-MacCamy model with logistic growth, we derive a numerical threshold parameter R_0^h , which ensures the global stability of the numerical trivial equilibrium whenever R_0^h<1 and the existence and the local stability of a unique numerical nontrivial equilibrium for R_0^h>1 . Moreover, under the presented partitioned implicit-explicit framework, the number R_0^h is also a threshold for the fully discrete scheme for any temporal stepsize. Finally, numerical experiments are presented to validate and demonstrate the efficiency of our theoretical results.
We introduce and analyze two Banach spaces-based new mixed finite element methods for a flow and transport model commonly encountered in sedimentation-consolidation processes, whose governing equations are given by the Brinkman flow with variable viscosity, coupled with a nonlinear advection–diffusion equation. The first variational formulation is based on a mixed approach for the Brinkman problem (written in terms of Cauchy stress and bulk velocity of the mixture) and the usual primal weak form for the transport equation. In turn, the second variational formulation arises from the introduction of the gradient of the solids volume fraction and the total (diffusive plus advective) flux for the concentration as new unknowns, which yields a momentum-conserving fully–mixed approach as the resulting system of equations. The respective continuous and discrete formulations are equivalently reformulated as fixed-point operator equations, whose solvability is established by combining the Schauder, Banach, and Brouwer theorems, with, among others, the Babuška–Brezzi theory and a recently introduced theory for perturbed saddle-point problems, both in Banach spaces, along with suitable regularity assumptions, Sobolev embeddings, and Rellich–Kondrachov compactness theorems. The mixed–primal and fully-mixed Galerkin schemes employ the classical Raviart–Thomas, piecewise continuous, and piecewise discontinuous polynomial approximations, for their corresponding unknowns. Next, Strang-type inequalities are utilized to rigorously derive optimal error estimates in the natural norms, which, combined with the approximation properties of the chosen finite element spaces yield optimal rates of convergence with respect to the mesh size. Finally, several numerical results illustrating the performance of both schemes and confirming the theoretical convergence rates are presented.
In this work, we analyze the numerical behavior of the classical Cauchy identity ∑ _λ s_λ (a_1,… ,a_n)s_λ (x_1,… ,x_m) = ∏ _j=1^n ∏ _i=1^m 1/1 - a_j x_i, by developing perturbation and running error analyses. We show that relative perturbations in the nodes x_i and coefficients a_j only induce small relative changes in the output provided some relative gaps are sufficiently large. We also propose an algorithm computing a posteriori relative error bound with low computational overhead. Finally, we derive truncation error bounds for the Schur expansion of the formula. Numerical experiments confirm the sharpness of the theoretical results and illustrate the effectiveness of the proposed bounds in practice.
