International Journal of Numerical Analysis and Modeling
International Journal of Numerical Analysis & Modeling (IJNAM) is directed to the broad spectrum of researchers in scientific computing, and publishes high quality papers in all fields of numerical analysis and mathematical modeling. IJNAM welcomes contribution of original developments of numerical methods and analysis leading to better understanding of the existing algorithms. Rigorous studies of the convergence, stability, and complexity for numerical methods are particularly appropriate for this journal.
Uniform Equivalence of $L^2$- and Discrete $ℓ^2$-Norms on $Q_1$-Finite Element Spaces with Mass Lumping in Any Finite Dimension
The $Q_1$-finite element spaces, in any finite $d$-dimension, are equipped with the discrete $ℓ^2_h$-inner product generated by the simple row-sum mass lumping. The equivalence of the discrete $ℓ^2_h$-norm and the $L^2$-norm on the $Q_1$-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 $h^1_h$-norm and $H^1$-norm. Numerical results are presented.
Copyright (c) 2026 International Journal of Numerical Analysis and Modeling
A High-Order Mixed Finite Element Method for Second Order Elliptic Equations on Curved Domain with Boundary Value Corrections
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 $(RT_k)$ of degree $k ≥ 1$ on triangular meshes, achieving an $O(h^{k+1/2})$ convergence in the $L^2$-norm estimate for the velocity field and an $O(h^k)$ convergence in the $H^1$-norm estimate for the pressure. Finally, numerical experiments validate our theoretical results.
Analysis of Transmission Line Model with Uncertain Parameter Using the PC-FDTD Method
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.
Analysis of Harmonic Average Method for Interface Problems with Discontinuous Solutions and Fluxes
Harmonic average method has been widely utilized to deal with heterogeneous coefficients in solving differential equations. One remarkable advantage of the harmonic averaging method is that no derivative of the coefficient is needed. Furthermore, the coefficient matrix of the finite difference equations is an M-matrix which guarantees the stability of the algorithm. It has been numerically observed but not theoretically proved that the method produces second order pointwise accuracy when the solution and flux are continuous even if the coefficient has finite discontinuities for which the method is inconsistent ($O$(1) in the local truncation errors). It has been believed that there are some fortunate error cancellations. The harmonic average method does not converge when the solution or the flux has finite discontinuities. In this paper, not only we rigorously prove the second order convergence of the harmonic averaging method for one-dimensional interface problem when the coefficient has a finite discontinuities and the solution and the flux are continuous, but also proposed an improved harmonic average method that is also second order accurate (in the $L^∞$ norm), which allows discontinuous solutions and fluxes along with the discontinuous coefficients. The key in the convergence proof is the construction of the Green’s function. The proof shows how the error cancellations occur in a subtle way. Numerical experiments in both 1D and 2D confirmed the theoretical proof of the improved harmonic average method.
An Efficient RVE-EMC Approach for Multiscale Equations of Random Heterogeneous Materials
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.
Enhanced Gradient Recovery-Based a Posteriori Error Estimator and Adaptive Finite Element Method for Elliptic Equations
Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators, one can not establish upper and lower a posteriori error bounds for the classical recovery type error estimators without the saturation assumption. In this paper, we first present three examples to show the unsatisfactory performance in the practice of standard residual or recovery-type error estimators, then, an improved gradient recovery-based a posteriori error estimator is constructed. The proposed error estimator contains two parts, one is the difference between the direct and post-processed gradient approximations, and the other is the residual of the recovered gradient. The reliability and efficiency of the enhanced estimator are derived. Based on the improved recovery-based error estimator and the newest-vertex bisection refinement method with a tailored mark strategy, an adaptive finite element algorithm is designed. We then prove the convergence of the adaptive method by establishing the contraction of gradient error plus oscillation. Numerical experiments are provided to illustrate the asymptotic exactness of the new recovery-based a posteriori error estimator and the high efficiency of the corresponding adaptive algorithm.
Combined Methods for Solving Time‑Varying Semilinear Differential‑Algebraic Equations with the Use of Spectral Projectors and Applications
Two combined methods for computing solutions of time-varying semilinear differential-algebraic equations (descriptor systems) are obtained. When constructing the methods, time-varying 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.
On a Priori Error Analysis of Discontinuous Galerkin Method for the Vlasov-Nonstationary Stokes System
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 Grönwall’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.
time-splitting scheme and numerical experiments
Copyright (c) 2025 International Journal of Numerical Analysis and Modeling
Physics-Based Multi-Time Stepping Algorithm for Three-Dimensional Quasi-Static Electroporoelasticity Equations
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.
A Fully Discrete Ultra-Weak Discontinuous Galerkin Method for Solving the Drift-Diffusion Model of Semiconductor Devices
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.
ultra-weak discontinuous Galerkin method
Analysis of Any-Order Runge-Kutta Spectral Volume Schemes for 1D Hyperbolic Equations
In this paper, we analyze any-order Runge-Kutta spectral volume schemes (RKSV(s,k)) for solving the one-dimensional scalar hyperbolic equation. The RKSV(s,k) was constructed by using the $s$-th explicit Runge-Kutta method in time-discretization which has strong-stability-preserving (SSP) property, and by letting a piecewise $k$−th degree ($k$ ≥ 1 is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with $k$ Gauss-Legendre points (LSV) or right-Radau points (RRSV). For the RKSV(s,k), we would like to establish a general framework which use the matrix transferring process technique for analyzing the stability and the convergence property. The framework for stability is evolved based on the energy equation, while the framework for error estimate is evolved based on the error equation. And the evolution process is represented by matrices. After the evolution is completed, three key indicative pieces of information are obtained: the termination factor $ζ$, the indicator factor $ρ$, and the final evolved matrix. We prove that for the RKSV(s,k), the stability holds and the $L_2$ norm error estimate is $\mathcal{O}(h^{k+1}+τ^s),$ provided that the CFL condition is satisfied. Our theoretical findings have been justified by several numerical experiments.
An Asymptotic Expansion-Based Deep Neural Network for Solving Singularly Perturbed Problems with Exponential Boundary Layers
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.
Auto-Stabilized Weak Galerkin Finite Element Methods for Biharmonic Equations on Polytopal Meshes Without Convexity Assumptions
This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for biharmonic equations with built-in stabilizers. Unlike existing stabilizer-free WG methods limited to convex elements in finite element partitions, our approach accommodates both convex and non-convex polytopal meshes, offering enhanced versatility. It employs bubble functions without the restrictive conditions required by existing stabilizer-free WG methods, thereby simplifying implementation and broadening application to various partial differential equations (PDEs). Additionally, our method supports flexible polynomial degrees in discretization and is applicable in any dimension, unlike existing stabilizer-free WG methods that are confined to specific polynomial degree combinations and 2D or 3D settings. We demonstrate optimal order error estimates for WG approximations in both a discrete $H^2$ norm for $k ≥ 2$ and an $L^2$ norm for $k > 2,$ as well as a sub-optimal error estimate in $L^2$ when $k = 2,$ where $k ≥ 2$ denotes the degree of polynomials in the approximation.
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A Unified Analysis Framework for Uniform Stability of Discretized Volterra Integrodifferential Equations
We provide a unified analysis framework for discretized Volterra integrodifferential equations by considering the $ϑ$-type convolution quadrature, where different $ϑ$ corresponds to different schemes. We first derive the long-time $l^∞$ stability of discrete solutions, and then prove a discrete Wiener-Lévy theorem to support the analysis of long-time $l^1$ 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.
Analysis of a Type II Thermal Problem Involving a Viscoelastic Beam
In this work, we will study, from both analytical and numerical points of view, a nonlocal problem involving a thermoviscoelastic 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.
Efficient and Long-Time Accurate Second-Order Decoupled Method for the Blood Solute Dynamics Model
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.
Provably Convergent Plug-and-Play Proximal Block Coordinate Descent Method for Hyperspectral Anomaly Detection
Hyperspectral anomaly detection refers to identifying pixels in the hyperspectral images that have spectral characteristics significantly different from the background. In this paper, we introduce a novel model that represents the background information using a low-rank representation. We integrate an implicit proximal denoiser prior, associated with a deep learning based denoiser, within a plug-and-play (PnP) framework to effectively remove noise from the eigenimages linked to the low-rank representation. Anomalies are characterized using a generalized group sparsity measure, denoted as $∥ · ∥_{2,ψ}.$ To solve the resulting orthogonal constrained nonconvex nonsmooth optimization problem, we develop a PnP-proximal block coordinate descent (PnP-PBCD) method, where the eigenimages are updated using a proximal denoiser within the PnP framework. We prove that any accumulation point of the sequence generated by the PnP-PBCD method is a stationary point. We evaluate the effectiveness of the PnP-PBCD method on hyperspectral anomaly detection in scenarios with and without Gaussian noise contamination. The results demonstrate that the proposed method can effectively detect anomalous objects, outperforming the competing methods that may mistakenly identify noise as anomalies or misidentify the anomalous objects due to noise interference.
Integrating Peridynamic Theory with Ground Fissure Mechanics: Application to Songzhuang Ground Fissure Study
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.
A Priori Error Analysis of the Local Discontinuous Galerkin Method for the Viscous Burgers-Poisson System
In this paper, we propose and analyze the local discontinuous Galerkin method for the viscous Burgers-Poisson system. The proposed method preserves two invariants and hence, yields solutions even for long time. A priori error estimates, which are of order $\mathcal{O}(h^{k+1})$, when polynomials of degree $k\geq 1$ are used for approximating solutions are established. Finally, numerical experiments are conducted to confirm our theoretical results.
New Finite Element Iterative Methods for Solving a Nonuniform Ionic Size Modified Poisson-Boltzmann Equation
In this paper, a nonuniform size modified Poisson-Boltzmann equation (SMPBE) for a protein in a solvent with multiple ionic species in distinct ionic sizes is derived by using a new electrostatic free energy functional and solution decomposition techniques. It is then proved to have a unique solution, and the solution satisfies a system consisting of nonlinear algebraic equations and one Poisson dielectric interface problem. To solve it numerically, two new finite element iterative schemes are proposed by using nonlinear successive over-relaxation techniques, along with an improved uniform SMPBE for generating initial iterates. Furthermore, they are programmed in Python and Fortran as a software package for solving the nonuniform SMPBE, and numerically tested on a Born ball test model and a protein in a sodium chloride solution and a sodium chloride and potassium chloride solution. Numerical results confirm the convergence of the two new iterative schemes and demonstrate the high performance of the new software package.
A Hybridizable Weak Galerkin Method for the Helmholtz Equation with Large Wave Number: $hp$ Analysis
In this paper, an $hp$ hybridizable weak Galerkin ($hp$-HWG) method is introduced to solve the Helmholtz equation with large wave number in two and three dimensions. By choosing a specific parameter and using the duality argument, we prove that the proposed method is stable under certain mesh constraint. Error estimate is obtained by using the stability analysis and the duality argument. Several numerical results are provided to confirm our theoretical results.
A 3D Conforming-Nonconforming Mixed Finite Element for Solving Symmetric Stress Stokes Equations
We propose a 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations. The low-order conforming finite elements are not inf-sup stable. The low-order nonconforming finite elements do not satisfy the Korn inequality. The proposed finite element space consists of two conforming components and one nonconforming component. We prove that the discrete inf-sup condition is valid and the discrete Korn inequality holds uniformly in the mesh-size. Based on these results we give some numerical verification. In addition, this element is compared numerically with six other mixed finite elements.
An Augmented IIM & Preconditioning Technique for Jump Embedded Boundary Conditions
A second-order accurate augmented method is proposed and analyzed in this paper for a general elliptic PDE with a general boundary condition using the jump embedded boundary conditions (JEBC) formulation. First of all, the existence and uniqueness of an interface problem with given are discussed. Then, the well-posedness theory is extended to the interface problems with given jump conditions. In the proposed numerical method, one novel idea is to preconditioning the PDE first so that the coefficient of the highest derivative is of $O(1)$. The second idea is to introduce two augmented variables corresponding to the jump in the solution and its normal derivative along the boundary to get an interface problem. For a piecewise constant coefficient, the fast Poisson solver then can be utilized in a rectangular domain. The augmented variables can be determined from a Schur complement system. We also propose two preconditioning techniques for the GMRES iterative method for the Schur complement; one is from the flux jump condition, and the other one is from the algebraic preconditioner based on the interpolation scheme in the augmented algorithm. The presented numerical results show that the proposed method has not only obtained second order accurate solutions in the $L^\u221e$\u00a0norm globally, but also second order accurate normal derivatives at the boundary from each side of the interface. The proposed preconditioning technique can speed up 50-90% compared with the method without preconditioning.
Jump embedded boundary conditions (JEBC)
A Nodal Sparse Grid Spectral Element Method for Multi-Dimensional Elliptic Partial Differential Equations
We develop a sparse grid spectral element method using nodal bases on Chebyshev-Gauss-Lobatto points for multi-dimensional elliptic equations. Since the quadratures based on sparse grid points do not have the accuracy of a usual Gauss quadrature, we construct the mass and stiffness matrices using a pseudo-spectral approach, which is exact for problems with constant coefficients and uniformly structured grids. Compared with the regular spectral element method, the proposed method has the flexibility of using a much less degree of freedom. In particular, we can use less points on edges to form a much smaller Schur-complement system with better conditioning. Preliminary error estimates and some numerical results are also presented.
A Simple Finite Element Method of the Cauchy Problem for Poisson Equation
In this paper, we introduce a simple method for the Cauchy problem. This new finite element method is based on least squares methodology with discontinuous approximations which can be implemented and analyzed easily. This discontinuous Galerkin finite element method is flexible to work with general unstructured meshes. Error estimates of the finite element solution are derived. The numerical examples are presented to demonstrate the robustness and flexibility of the proposed method.
Maximal Lp Error Analysis of FEMs for Nonlinear Parabolic Equations with Nonsmooth Coefficients
The paper is concerned with $L^p$ error analysis of semi-discrete Galerkin FEMs for nonlinear parabolic equations. The classical energy approach relies heavily on the strong regularity assumption of the diffusion coefficient, which may not be satisfied in many physical applications. Here we focus our attention on a general nonlinear parabolic equation (or system) in a convex polygon or polyhedron with a nonlinear and Lipschitz continuous diffusion coefficient. We first establish the discrete maximal $L^p$-regularity for a linear parabolic equation with time-dependent diffusion coefficients in $L^\u221e(0,T;W^{1,N+\epsilon}) \cap C(\overline{\Omega} \times [0,T])$ for some $\epsilon>0$, where $N$ denotes the dimension of the domain, while previous analyses were restricted to the problem with certain stronger regularity assumption. With the proved discrete maximal $L^p$-regularity, we then establish an optimal $L^p$ error estimate and an almost optimal $L^\u221e$ error estimate of the finite element solution for the nonlinear parabolic equation.
High Degree Immersed Finite Element Spaces by a Least Squares Method
We present a least squares framework for constructing $p$-th degree immersed finite element (IFE) spaces for typical second-order elliptic interface problems. This least squares formulation enforces interface jump conditions including extended ones already proposed in the literature, and it guarantees the existence of $p$-th IFE shape functions on interface elements. The uniqueness of the proposed $p$-th degree IFE shape functions is also discussed. Computational results are presented to demonstrate the approximation capabilities of the proposed $p$-th IFE spaces as well as other features.
Stochastic Spline-Collocation Method for Constrained Optimal Control Problem Governed by Random Elliptic PDE
In this paper, we investigate a stochastic spline-collocation approximation scheme for an optimal control problem governed by an elliptic PDE with random field coefficients. We obtain the necessary and sufficient optimality conditions for the optimal control problem and establish a scheme to approximate the optimality system through the discretization with respect to the spatial space by finite elements method and the probability space by stochastic spline-collocation method. We further investigate Smolyak approximation schemes, which are effective collocation strategies for smooth problems that depend on a moderately large number of random variables. For more general control problems where the state may be non-smooth with respect to the random variables in some areas, we adopt a domain decomposition strategy to partition the random space into smooth and non-smooth parts and then apply Smolyak scheme and spline approximation respectively. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.
A Finite Element Method for the One-Dimensional Prescribed Curvature Problem
We develop a finite element method for solving the Dirichlet problem of the one- dimensional prescribed curvature equation due to its irreplaceable role in applications. Specifically, we first analyze the existence and uniqueness of the solution of the problem and then develop a finite element method to solve it. The well-posedness of the finite element method is shown by employing the Banach fixed-point theorem. The optimal error estimates of the proposed method in both the $H^1$ norm and the $L^2$ norm are established. We also design a Newton type iteration scheme to solve the resulting discrete nonlinear system. Numerical experiments are presented to confirm the order of convergence of the proposed method.
A Posteriori Error Estimates for Mixed Finite Element Galerkin Approximations to Second Order Linear Hyperbolic Equations
In this article, a posteriori error analysis for mixed finite element Galerkin approximations of second order linear hyperbolic equations is discussed. Based on mixed elliptic reconstructions and an integration tool, which is a variation of Baker's technique introduced earlier by G. Baker (SIAM J. Numer. Anal., 13 (1976), 564-576) in the context of a priori estimates for a second order wave equation, a posteriori error estimates of the displacement in $L^\u221e(L^2)$-norm for the semidiscrete scheme are derived. Finally, a first order implicit-in-time discrete scheme is analyzed and a posteriori error estimators are established.
first order implicit completely discrete scheme