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.
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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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.
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.
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.
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.
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.
Nattapol Ploymaklam, Pratik M. Kumbhar & Amiya K. Pani
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.
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.
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 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.
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 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.
Susanne C. Brenner, Li-Yeng Sung, Zhuo Wang & Yuesheng Xu
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.
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.
Benxue Gong, Liang Ge, Tongjun Sun, Wanfang Shen & W.B. Liu
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.
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.
A Third Order Linearized BDF Scheme for Maxwell's Equations with Nonlinear Conductivity Using Finite Element Method
In this paper, we study a third order accurate linearized backward differential formula (BDF) type scheme for the nonlinear Maxwell's equations, using the N\u00e9delec finite element approximation in space. A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. An optimal $L^2$ error estimate is presented, via a linearized stability analysis for the numerical error function, under a condition for the time step, $\tau \leq C^*_0h^2$ for a fixed constant $C^*_0$. Numerical results are provided to confirm our theoretical analysis and demonstrate the high order accuracy and stability (convergence) of the linearized BDF finite element method.
Maxwell's equations with nonlinear conductivity
convergence analysis and optimal error estimate
Changhui Yao, Yanping Lin, Cheng Wang & Yanli Kou
A New Finite Element for Interface Problems Having Robin Type Jump
We propose a new finite element method for solving second order elliptic interface problems whose solution has a Robin type jump along the interface. We cast the problem into a new variational form and introduce a finite element method to solve it using a uniform grid. We modify the $P_1$-Crouzeix-Raviart element so that the shape functions satisfy the jump conditions along the interface. We note that there are cases that the Lagrange type basis can not be used because of the jump in the value. Numerical experiments are provided.
$P_1$-Crouzeix-Raviart finite element method.
Do Y. Kwak, Seungwoo Lee & Yunkyong Hyon
Finite Element Method to Control the Domain Singularities of Poisson Equation Using the Stress Intensity Factor: Mixed Boundary Condition
In this article, we consider the Poisson equation on a polygonal domain with the domain singularity raised from the changed boundary conditions with the inner angle $\omega>\frac{\pi}{2}$. The solution of the Poisson equation with such singularity has a singular decomposition: regular part plus singular part. The singular part is a linear combination of one or two singular functions. The coefficients of the singular functions are usually called stress intensity factors and can be computed by the extraction formula. In [11] we introduced a new partial differential equation which has 'zero' stress intensity factor using this stress intensity factor, from whose solution we can obtain a very accurate solution of the original problem simply by adding singular part. Although the method in [11] works well for the Poisson problem with Dirichlet boundary condition, it does not give optimal results for the case with stronger singularity, for example, mixed boundary condition with bigger inner angle. In this paper we give a revised algorithm which gives optimal convergences for both cases.
Analysis of an Embedded Discontinuous Galerkin Method with Implicit-Explicit Time-Marching for Convection-Diffusion Problems
In this paper, we analyze implicit-explicit (IMEX) Runge-Kutta (RK) time discretization methods for solving linear convection-diffusion equations. The diffusion operator is treated implicitly via the embedded discontinuous Galerkin (EDG) method and the convection operator explicitly via the upwinding discontinuous Galerkin method.
implicit-explicit Runge-Kutta time-marching scheme
Overlapping Domain Decomposition Preconditioners for Unconstrained Elliptic Optimal Control Problems
In this paper, we propose several overlapping domain decomposition preconditioners for solving the unconstrained elliptic optimal control problem, based on the two level additive Schwarz algorithm. We consider the cases with controls on the whole domain and controls from a local subset. The latter case can be viewed as the subproblems when we solve the control-constrained control problem by using semi-smooth Newton method. When the controls act on the whole domain, we construct a symmetric and positive definite preconditioner which is proved to be robust combined with preconditioned MINRES method, and a symmetric and indefinite preconditioner which can be used in the preconditioned GMRES method and shows better numerical performance than the positive definite one. When the controls act on a local subset, we also construct a similar symmetric and indefinite preconditioner, the numerical experiments show its efficiency when combined with preconditioned GMRES 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
An Efficient Multigrid Method for Ground State Solution of Bose-Einstein Condensates
An efficient multigrid method is proposed to compute the ground state solution of\r Bose-Einstein condensations by the finite element method based on the combination of the multigrid method for nonlinear eigenvalue problem and an efficient implementation for the nonlinear\r iteration. The proposed numerical method not only has the optimal convergence rate, but also has\r the asymptotically optimal computational efficiency which is independent from the nonlinearity\r of the problem. The independence from the nonlinearity means that the asymptotic estimate of\r the computational work can reach almost the same as that of solving the corresponding linear\r boundary value problem by the multigrid method. Some numerical experiments are provided to\r validate the efficiency of the proposed method.
Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes
We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite\r element method (DGFEM) for the numerical approximation of the biharmonic equation on general\r computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the\r stability and $hp$-version a-priori error bound are derived based on the specific choice of the interior\r penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new\r inverse inequality for a special class of polynomial functions (harmonic polynomials), the proposed\r DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with\r an $arbitrary$ number of faces for polynomial basis with degree $p$ = 2, 3. The key feature of the\r proposed method is that it employs elemental polynomial bases of total degree $\mathcal{P}$$p$, defined in the\r physical coordinate system, without requiring the mapping from a given reference or canonical\r frame. A series of numerical experiments are presented to demonstrate the performance of the\r proposed DGFEM on general polygonal/polyhedral meshes.
A Posteriori Error Analysis of an Augmented Dual-Mixed Method in Linear Elasticity with Mixed Boundary Conditions
We consider an augmented mixed finite element method for the equations of plane\r linear elasticity with mixed boundary conditions. The method provides simultaneous approximations of the displacements, the stress tensor and the rotation. We develop an a posteriori error\r analysis based on the Ritz projection of the error and the use of an appropriate auxiliary function, and derive fully local reliable a posteriori error estimates that are locally efficient up to the\r elements that touch the Neumann boundary. We provide numerical experiments that illustrate\r the performance of the corresponding adaptive algorithm and support its use in practice.
Robust and Efficient Mixed Hybrid Discontinuous Finite Element Methods for Elliptic Interface Problems
Because of the discontinuity of the interface problems, it is natural to apply the\r discontinuous Galerkin (DG) finite element methods to solve those problems. In this work, both\r fitted and unfitted mixed hybrid discontinuous Galerkin (MHDG) finite element methods are\r proposed to solve the elliptic interface problems. For the fitted case, the problems can be solved\r directly by MHDG method. For the unfitted case, the broken basis functions (unnecessary to\r satisfy the jump conditions) are introduced to those elements which are cut across by interface,\r the weights depending on the volume fractions of cut elements and the different diffusions (or\r material heterogeneities) are used to stabilize the method, and the idea of the Nitsche's penalty\r method is applied to guarantee the jumps on the interface parts of cut elements. Unlike the\r immersed interface finite element methods (IIFEM), the two jump conditions are enforced weakly\r in our variational formulations. So, our unfitted interface MHDG method can be applied more\r easily than IIFEM to general cases, particularly when the immersed basis function cannot be\r constructed. Numerical results on convergence and sensitivities of both interface location within a\r cut element and material heterogeneities show that the proposed methods are robust and efficient\r for interface problems.
discontinuous Galerkin finite element methods
sensitivities of interface location and material heterogeneities.
A Relaxation Approach to Discretization of Boundary Optimal Control Problems of Semilinear Parabolic Equations
We consider an optimal boundary control problem described by a semilinear parabolic\r partial differential equation, with control and state constraints. Since this problem may have no\r classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized\r by using a finite element method in space and a partially implicit scheme in time, while the\r controls are approximated by piecewise constant relaxed controls. We first state the necessary\r conditions for optimality for the continuous problem and the discrete relaxed problem. Next,\r under appropriate assumptions, we prove that accumulation points of sequences of optimal (resp.\r admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal)\r for the continuous relaxed problem.
Anisotropic Mesh Adaptation Method Based on Anisotropic Bubble-Type Local Mesh Generation
A new anisotropic adaptive mesh refinement method based on anisotropic bubble-type local mesh generation (ABLMG) for elliptic partial differential equations is proposed. The anisotropic meshes are generated as quasi-uniform meshes in metric spaces with the metric determined on each vertex by anisotropic a posteriori error estimator. Under the new metric, the error is equidistributed in the directions of maximum and minimum stretching on an element, and the mesh size is reduced/coarsened in regions with large/small errors. With the full use of the adjacent lists provided by the node placement method, the local mesh for each vertex is generated through ABLMG method. Compared with other methods, the mesh refining and coarsening can be obtained in the same framework and the mesh suits the metric well at each refinement level. Numerical results in two-dimensions are presented to verify the ability of our metric tensor to generate anisotropic mesh with correct concentration and stretching direction.
node placement and local mesh generation.
Postprocessing of Continuous Galerkin Solutions for Delay Differential Equations with Nonlinear Vanishing Delay
In this paper we propose several postprocessing techniques to accelerate the convergence of the continuous Galerkin solutions for delay differential equations with nonlinear vanishing delay. They are interpolation postprocessings (including integration type, Lagrange type, and\r polynomial preserving recovery type) and iteration postprocessing. The theoretical expectations\r are confirmed by numerical experiments.