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This paper proposes a difference finite element method (DFEM) for solving the Poisson equation in the four-dimensional (4D) domain Ω. The method combines finite difference discretization based on the Q 1 -element in the third and fourth directions with finite element discretization based on the P 1 -element in the other directions. In this way, the numerical solution of the 4D Poisson equation can be transformed into a series of finite element solutions of the 2D Poisson equation. Moreover, we prove that the DFE solution u H satisfies H 1 -stability, and the error function u H − u achieves first-order convergence under the H 1 -error. Finally, we provide three numerical examples to verify the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

This paper deals with the temperature profile of a simple combustion and presents the alternative exact formulas for the temperature profile of the planar vessel. The differential equation that describes this system is referred as a Bratu equation or Poisson's equation in one-dimensional steady state case. In this present study, new solutions with general boundary conditions are developed. The results are compared with numerical solutions using Maxima, a computer algebra system program capable of numerical and symbolic computation. The new solutions yield formula that may provide a valuable information about relationship between terms, variables and coefficients which can be useful for theoretical physics. [ABSTRACT FROM AUTHOR]

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian (− Δ) α 2 . Numerical analysis and experiments are provided to study its performance. Our method has the same symbol | ξ | α as the fractional Laplacian (− Δ) α 2 at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This unique feature distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark 1.1). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is O (2 N log ⁡ (2 N)) , and the memory storage is O (N) with N the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems. [ABSTRACT FROM AUTHOR]

This paper explores the utilization of randomized SVD (rSVD) in the context of kernel matrices arising from radial basis functions (RBFs) for the purpose of solving interpolation and Poisson problems. We propose a truncated version of rSVD, called trSVD, which yields a stable solution with a reduced condition number in comparison to the non-truncated variant, particularly when manipulating the scale or shape parameter of RBFs. Notably, trSVD exhibits exceptional proficiency in capturing the most significant singular values, enabling the extraction of critical information from the data. When compared to the conventional truncated SVD (tSVD), trSVD achieves comparable accuracy while demonstrating improved efficiency. Furthermore, we explore the potential of trSVD by employing scale parameter strategies, such as leave-one-out cross-validation and effective condition number. Then, we apply trSVD to solve a 2D Poisson equation, thereby showcasing its efficacy in handling partial differential equations. In summary, this study offers an efficient and accurate solver for RBF problems, demonstrating its practical applicability. The code implementation is provided to the scientific community for their access and reference. [ABSTRACT FROM AUTHOR]

For the spatial discretization of elliptic and parabolic partial differential equations (PDEs), we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order k ∈ N. On a quite general class of 2D domains, namely separable domains , and even on special surfaces, the discrete problem is then reformulated as a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. By considering the IMEX Euler method for the PDE time discretization, we obtain a sequence of these equations. To solve each multiterm Sylvester equation, we apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method with a matrix-oriented preconditioner that captures the spectral properties of the Sylvester operator. Solving the Poisson problem and the heat equation on some separable domains by MO-FEM-PCG, we show a gain in computational time and memory occupation wrt the classical vector PCG with same preconditioning or wrt a LU based direct method. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns on some separable domains and cylindrical surfaces for a morphochemical reaction-diffusion PDE system for battery modelling. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for incompressible fluid flow. A difficulty in analyzing incompressible fluid flow is that the continuity equation has no time evolution term. In the marker and cell (MAC) method, Poisson's equation is solved iteratively, which takes most of the computation time, and in the artificial compressibility method (ACM), pseudo-time iteration is necessary to solve for unsteady solutions. Here, modal analysis that uses the velocity eigenvectors corresponding to zero eigenvalues is proposed for analyzing two-dimensional incompressible fluid flow. The proposed method involves only about one third of the number of variables needed in the MAC method and the ACM, and it does not require iterative calculation of Poisson's equation or pseudo-time iteration. Numerical results for a simple flow system and a cavity flow obtained using the proposed method are compared with those obtained using the ACM and the simplified MAC method. The results agree well, thereby validating the proposed modal analysis. [ABSTRACT FROM AUTHOR]

Efficient solution of the Vlasov equation, which can be up to six-dimensional, is key to the simulation of many difficult problems in plasma physics. The discontinuous Petrov-Galerkin (DPG) finite element methodology provides a framework for the development of stable (in the sense of Ladyzhenskaya–Babuška–Brezzi conditions) finite element formulations, with built-in mechanisms for adaptivity. While DPG has been studied extensively in the context of steady-state problems and to a lesser extent with space-time discretizations of transient problems, relatively little attention has been paid to time-marching approaches. In the present work, we study a first application of time-marching DPG to the Vlasov equation, using backward Euler for a Vlasov-Poisson discretization. We demonstrate adaptive mesh refinement for two problems: the two-stream instability problem, and a cold diode problem. We believe the present work is novel both in its application of unstructured adaptive mesh refinement (as opposed to block-structured adaptivity, which has been studied previously) in the context of Vlasov-Poisson, as well as in its application of DPG to the Vlasov-Poisson system. We also discuss extensive additions to the Camellia library in support of both the present formulation as well as extensions to higher dimensions, Maxwell equations, and space-time formulations. [ABSTRACT FROM AUTHOR]

This paper proposes a novel meshless local integral equation (LIE) method for numerical solutions of two and three-dimensional Poisson equations. The proposed method can be regarded as a new variant of the meshless local Petrov-Galerkin (MLPG) method which already has six variants, and so the proposed method can be called MLPG7. A variant of MLPG called local boundary integral equation (LBIE) method, uses a localized fundamental solution of Laplace equation as test function. This test function vanishes on local boundaries and therefore, the derived local equations do not involve the gradient of field variables. The motivation for new formulation is using an average of LBIE equation over radius of local sub-domains instead of a single sub-domain. A new kernel is introduced which is a new modification of fundamental solutions of Laplace equation. It is proved that using the new kernel as test function leads to the same local equations as averaging. Infact, the new formulation is a modification of the LBIE method for which the boundary integral is replaced with a domain integral. It is proved that new local weak formulation is equivalent with strong form. The convergence of the method for the case of regularly spaced nodal points is proved. For trial approximation, this paper uses Gaussian radial basis functions (Gaussian RBF), however, derivative free property of the method allows the use of any non-differentiable basis functions. Stability and convergence of the method are numerically studied by considering both 2D and 3D problems with regular and irregular nodal points. The method is compared with LBIE, finite differences (FD) and RBF-FD methods and the numerical results reveal the significance of the proposed method. [ABSTRACT FROM AUTHOR]

In this study, a new Poisson-type equation for non-hydrostatic pressure in terrain-following σ -coordinates is proposed. The resulting equation consists of a standard elliptic operator and other first-order operators for non-hydrostatic pressure. Compared with the original Poisson-type equation, the treatment of the boundary condition at the lateral boundary can be simplified. Numerical tests indicate that the reformulated Poisson-type equation shows better convergence properties than the original equation when the quadrature-free nodal discontinuous Galerkin method is adopted for numerical discretization. Additionally, for the standing wave case, the numerical error of the three-dimensional non-hydrostatic model with the reformulated Poisson equation is smaller than that with the original equation, the numerical results of the solitary wave case indicate that the dissipation of the three-dimensional non-hydrostatic model is small, and transformation of wave over a circular shoal indicates that the developed model can properly simulate combined wave refraction and diffraction. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel discontinuous Galerkin (DG) finite element method for solving the Poisson's equation u x x + u y y = f (x , y) on Cartesian grids. The proposed method consists of first applying the standard DG method in the x -spatial variable leading to a system of ordinary differential equations (ODEs) in the y -variable. Then, using the method of line, the DG method is directly applied to discretize the resulting system of ODEs. In fact, we propose a fully DG scheme that uses p -th and q -th degree DG methods in the x and y variables, respectively. We show that, under proper choices of numerical fluxes, the method achieves optimal convergence rate in the L 2 -norm of O (h p + 1) + O (k q + 1) for the DG solution, where h and k denote, respectively, the mesh step sizes for the x and y variables. Our theoretical results are validated through several numerical experiments. [ABSTRACT FROM AUTHOR]

In this study, we use the log-linear link function and propose a generalized fused Lasso (GFL) Poisson regression model in which the nonlinear trend is discretely represented by categorical covariates in the additive model. We use the coordinate descent algorithm for the estimation and show that the optimal solution in a coordinate axis can be found explicitly. To demonstrate the proposed approach, we analyze Japanese crime data. Simulation results showed a fitness ratio for true fusion to be more than 90% in total, demonstrating the reliability of the estimates. [ABSTRACT FROM AUTHOR]

We focus on the ill posed data completion problem and its finite element approximation, when recast via the variational duplication Kohn–Vogelius artifice and the condensation Steklov–Poincaré operators. We try to understand the useful hidden features of both exact and discrete problems. When discretized with finite elements of degree one, the discrete and exact problems behave in diametrically opposite ways. Indeed, existence of the discrete solution is always guaranteed while its uniqueness may be lost. In contrast, the solution of the exact problem may not exist, but it is unique. We show how existence of the so called " weak spurious modes ", of the exact variational formulation, is source of instability and the reason why existence may fail. For the discrete problem, we find that the cause of non uniqueness is actually the occurrence of " spurious modes ". We track their fading effect asymptotically when the mesh size tends to zero. In order to restore uniqueness, we recall the discrete version of the Holmgren principle, introduced in Azaïez et al. (2011) [4] , and we discuss the effect on uniqueness of the finite element mesh, using some graph theory basic material. [ABSTRACT FROM AUTHOR]

We investigate a potential model for an unbounded celestial bodies of finite mass composed of a solid core and a gaseous atmosphere. The system is governed by the Navier–Stokes–Fourier–Poisson equations, incorporating no-slip boundary conditions for velocity and a specified temperature distribution on the surface of the solid core. Additionally, a positive far-field condition is imposed on the temperature. This manuscript extends the mathematical theory of open fluid systems to unbounded exterior domains addressing these physically motivated yet highly challenging combination of boundary conditions. Notably, we establish the existence of global-in-time weak solutions and demonstrate the weak–strong uniqueness principle [ABSTRACT FROM AUTHOR]

Silicon carbide(SiC) double gate junctionless metal oxide semiconductor field-effect transistors(DG JL MOSFETs) have attracted significant attention due to their ideal high temperature characteristics and radiation resistance. Therefore, it is meaningful to exploit an I-V model for SiC DG JL MOSFETs. In this article, we make a linear approximation to describe the relationship between the surface mobile charge density and the surface electron concentration of the device. Based on this approximation and using the one-dimensional Poisson's equation, we solve for the potential distribution of a SiC DG JL MOSFET in the subthreshold region. From this solution, we derived a functional relationship between the surface mobile charge density in the channel and the channel quasi-Fermi potential. Then we successfully developed a unified I-V model for the SiC DG JL MOSFETs. Based on the drain to source current calculation formula, the calculation expressions for the device's transconductance and output conductance are derived. By comparing our model with the results from the two-dimensional numerical simulation software Silvaco Atlas, our model's calculations closely match the two-dimensional numerical simulation results from the subthreshold region to the accumulation region. This model has reference significance for SiC DG JL MOSFETs in the high temperature electronic circuit application field. [ABSTRACT FROM AUTHOR]

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections. • A highly accurate and efficient non-reflection nonlinear numerical wave tank has been established by the incorporation of the Harmonic Polynomial Cell (HPC) method with a mass source wave generation method. • The relationship between the internal source function and the expected waveform is investigated. The simulation covered the propagation of various types of waves, including second-order Stokes waves, fifth-order Stokes waves, solitary waves, fifth-order Fenton stream function waves, random waves, and the stationary wave. • The non-reflection characteristics of the HPC-MS method in the region of the wave-making source are investigated by arranging a fully reflective vertical wall. • The numerical tank based on the present HPC-MS method can be employed as a tool for long-time simulation of wave-body interaction, avoiding the effect of second reflection of the propagating wave due to the reflection from the front edge of the body reaching the wave generation boundary again. [ABSTRACT FROM AUTHOR]

In this paper, we propose a locking-free numerical method to solve the quasi-static linear thermo-poroelasticity model, which exhibits two types of locking phenomena: Poisson locking and nonphysical oscillations. By analyzing the regularity of solution of the original model, we find that the Poisson locking is caused by div u ≈ 0 as λ → + ∞. Moreover, when discretizing the diffusion equations using backward Euler method for time, one can see that div u 1 ≈ 0 for some special parameters. If we directly use the continuous Galerkin mixed finite element method (FEM) to solve the original model, the pressure and temperature fields exhibit numerical oscillations at early times. To overcome these two locking phenomena, we introduce a new variable ξ = α p + β T − λ div u to reformulate the original problem into a new one. This new problem includes a built-in mechanism to maintain stability for the continuous Galerkin mixed FEM. We further prove the existence and uniqueness of the weak solution by using the standard Galerkin method in conjunction with regularity estimates. We also design a fully discrete time-stepping scheme that employs mixed FEM with P 2 − P 1 − P 1 − P 1 element pairs for the space variables and the backward Euler method for the time variable. Optimal convergence is demonstrated in both space and time. Finally, numerical examples are provided to demonstrate the optimal convergence rates of the variables and the robustness of the proposed method with respect to ν , and to verify the absence of the locking phenomenon. [ABSTRACT FROM AUTHOR]

In a recent paper, a novel coding of the Ghost Fluid Method for the variable coefficient Poisson equation with discontinuous functions (named GFMxP) was proposed. A lot of numerical tests, with all the required quantities available in a analytic form, were used to demonstrate the ability of the new procedure in modeling a sharp interface and to check the accuracy order of the solutions. In practical applications, however, the real difficulty stands in the estimation of the so-called "ghost values", that is the values at points where the function is not only unknown, but even not defined. These values allow to compute the corrective terms enabling the use of standard finite difference formulas in presence of a singularity and/or a discontinuity, and can be only determined through some extrapolation procedure, whose truthfulness is essential to achieve a reliable result. The paper deals with such a basic issue, by testing different numerical strategies and demonstrating the strict relationship between the order of the adopted fit-model, the order of the solving scheme for the Poisson equation and the accuracy of the final solution. [Display omitted] • The paper deals with a GFMxP-based solution of the Poisson equation and the extrapolation of the ghost values. • Both a polynomial fit and a PDE-based extrapolation technique are used. • A mathematical and numerical comparison between the Aslam and Bochkov–Gibou procedures is proposed. • The effectiveness of the PDE fit model is demonstrated by a lot of tests, equipped with detailed error analyses. • The presented results could be relevant for a lot of applications of practical interest. [ABSTRACT FROM AUTHOR]

In this survey, we describe controlled interacting particle systems (CIPS) to approximate the solution of the optimal filtering and the optimal control problems. Part I of the survey is focussed on the feedback particle filter (FPF) algorithm, its derivation based on optimal transportation theory, and its relationship to the ensemble Kalman filter (EnKF) and the conventional sequential importance sampling–resampling (SIR) particle filters. The central numerical problem of FPF—to approximate the solution of the Poisson equation—is described together with the main solution approaches. An analytical and numerical comparison with the SIR particle filter is given to illustrate the advantages of the CIPS approach. Part II of the survey is focussed on adapting these algorithms for the problem of reinforcement learning. The survey includes several remarks that describe extensions as well as open problems in this subject. • A tutorial style survey of the feedback particle filter (FPF) algorithm. • Relationship to optimal transportation theory. • Relationship to recent developments in data assimilation and reinforcement learning. [ABSTRACT FROM AUTHOR]

• Meshfree hybrid discretization of the diffusion operator with discontinuous coefficients. • Formulation of a conservative scheme to enforce Neumann boundary conditions. • Comparison of the introduced methods with respect to convergence and performance depending on jump magnitude. We present a meshfree generalized finite difference method for solving Poisson's equation with a diffusion coefficient that contains jump discontinuities up to several orders of magnitude. To discretize the diffusion operator, we formulate a strong form method that uses a smearing of the discontinuity; and a conservative formulation based on locally computed Voronoi cells. Additionally, we propose a novel conservative formulation for enforcing Neumann boundary conditions that is compatible with the conservative formulation of the diffusion operator. Finally, we introduce a way to switch from the strong form to the conservative formulation to obtain a locally conservative and positivity preserving scheme. The presented numerical methods are benchmarked against four test cases of varying complexity and jump magnitude on point clouds with nodes that are not aligned to the discontinuity. Our results show that the new hybrid method that switches between the two formulations produces better results than the classical generalized finite difference approach for high jumps in diffusivity. [ABSTRACT FROM AUTHOR]

We present a high order finite difference solver, ParaTC, for the direct numerical simulations of turbidity currents with canonical turbulent channel configuration (periodic boundary conditions in the horizontal directions and non-periodic in vertical direction). Uniform meshes are adopted in streamwise and spanwise directions, while stretched grids can be used in wall-normal direction. In order to improve the parallel efficiency, we propose a new 2D pencil-like parallel configuration with totally 6 different pencil arrangements. A parallel Thomas algorithm is also included to further reduce the communication overhead when solving tridiagonal equations. In addition, we perform an optimal search method in the initializing stage to find the fastest Poisson solver scheme among four alternatives for a specific mesh configuration. The runtime ratio between traditional pencil-like Poisson solver and present solver is about 1.5. An approximate linear strong scaling performance is achieved, and the weak scaling performance is also improved. Three benchmark simulations are preformed, and the statistics are compared with those extracted from the simulations by the spectral method, and good agreements are achieved. The source code is freely available at https://github.com/GongZheng-Justin/ParaTC. [ABSTRACT FROM AUTHOR]

Meshless methods have become an essential numerical tool for simulating a wide range of flow–structure interaction problems. However, the way by which the impermeable solid boundary condition is implemented can significantly affect the accuracy of the results and computational cost. This paper develops an improved boundary scheme through a weak formulation for the boundary particles based on Pressure Poisson's Equation (PPE). In this scheme, the wall boundary particles simultaneously satisfy the PPE in the local integration domain by adopting the Meshless Local Petrov–Galerkin method with the Rankine source solution (MLPG_R) integration scheme (Ma, 2005b) and the Neumann boundary condition, i.e., normal pressure gradient condition, on the wall boundary which truncates the local integration domain. The new weak formulation vanishes the derivatives of the unknown pressure at wall particles and is discretized in the truncated support domain without extra artificial treatment. This improved boundary scheme is validated by analytical solutions, numerical benchmarks, and experimental data in the cases of patch tests, lid-driven cavity, flow over a cylinder and monochromic wave generation. Second-order convergent rate is achieved even for disordered particle distributions. The results show higher accuracy in pressure and velocity, especially near the boundary, compared to the existing boundary treatment methods that directly discretize the pressure Neumann boundary condition. [ABSTRACT FROM AUTHOR]

In this work, we explore the numerical solution of geometric shape optimization problems using neural network-based approaches. This involves minimizing a numerical criterion that includes solving a partial differential equation with respect to a domain, often under geometric constraints like a constant volume. We successfully develop a proof of concept using a flexible and parallelizable methodology to tackle these problems. We focus on a prototypal problem: minimizing the so-called Dirichlet energy with respect to the domain under a volume constraint, involving Poisson's equation in R 2. We use variational neural networks to approximate the solution to Poisson's equation on a given domain, and represent the shape through a neural network that approximates a volume-preserving transformation from an initial shape to an optimal one. These processes are combined in a single optimization algorithm that minimizes the Dirichlet energy. A significant advantage of this approach is its inherent parallelizability, which makes it easy to handle the addition of parameters. Additionally, it does not rely on shape derivative or adjoint calculations. Our approach is tested on Dirichlet and Robin boundary conditions, parametric right-hand sides, and extended to Bernoulli-type free boundary problems. The source code for solving the shape optimization problem is open-source and freely available. [ABSTRACT FROM AUTHOR]

The electron properties of two coupled quantum wells with a δ doping layer are investigated by solving simultaneous Schrödinger and Poisson equations while subjecting the nanosystem to a magnetic field ranging from 0 T to 30 T. The calculation is conducted using the FEniCS Project and Python programming within the effective mass and Hartree approximations. The presence of a magnetic field significantly modifies the electron density, a factor not previously considered, making this the first work to address the modification of the density of states of an electron system by a magnetic field and showing the formation of Landau levels in quantum wells. We demonstrate that the magnetic field significantly alters the electron density, Landau levels, and Fermi energy. Furthermore, these physical parameters are influenced by the dimensions of the nanostructure, represented by L d and L b , as well as the properties of the layers, such as δ , n d , and x i. • We proposes a modulation of n-doped GaAs/AlGaAs coupled quantum well under a magnetic field effect, based on the simultaneous solution of Schrödinger's and Poisson's equations. • An increasing in the magnetic field causes the electron wavefunction to enclose the doped region, emphasizing tunneling phenomena and enhancing the coupling between quantum wells. • The Fermi energy initially decreases with low magnetic fields and shifts from the conduction band to the forbidden gap of GaAs at around 2 – 3 T. • The Fermi energy and Landau levels gradually decrease as the well size expands, independent of the doping barrier center's position or doping in the right quantum well. • Electron density decreases with respect to Landau levels until it reaches zero under a uniform magnetic field. This behavior varies with L d , n d , L b , and δ under the magnetic field. [ABSTRACT FROM AUTHOR]

We propose in this paper two kinds of generalized scalar auxiliary variable (GSAV) approach with relaxation (R-GSAV) for the Navier–Stokes–Poisson–Nernst–Planck equations. By applying positive function transform approach for ion concentration equation, introducing auxiliary variable for energy equation and adopting consistent splitting approach for momentum equations, we construct fully decoupled, linearized and k th-order semi-implicit schemes. The proposed schemes have several advantages: the concentration components of the discrete solution preserve the properties of positivity and mass conservation; these are unconditionally energy stable; the corrected energy is consistent with the original energy; only require solving decoupled linear systems with constant coefficients at each time step. We present some numerical results to validate these schemes and investigate the dynamics with initial discontinuous concentrations. • Two kinds of generalized scalar auxiliary variable (GSAV) approach with relaxation (R- GSAV) are proposed for the Navier–Stokes–Poisson–Nernst–Planck equations. • By applying positive function transform approach for ion concentration equation, introducing auxiliary variable for energy equation and adopting consistent splitting approach for momentum equations, fully decoupled, linearized and k th order semi-implicit schemes are proposed. • The proposed schemes have several advantages: the concentration components of the discrete solution preserve the properties of positivity and mass conservation; these are unconditionally energy stable; the corrected energy is consistent with the original energy. • Only require solving decoupled linear systems with constant coefficients at each time step. • Numerical results to validate these schemes and investigate the dynamics with initial discontinuous concentrations are presented. [ABSTRACT FROM AUTHOR]

In order to modeling of junctionless (JL) surrounding-gate (SG) nanosheet MOSFET more accurately, a new model for determining threshold voltage and drain-induced barrier lowering (DIBL) effect of JL SG nanosheet NMOSFET is proposed through deriving the Poisson's equation under rectangular coordinate system. The model captures quantum confinement effect and source/drain depletion regions, it is validated through the Sentaurus TCAD simulation results. Variations of source/drain depletion regions with the channel width, height, doping concentration, the gate bias, the drain bias and variations of threshold voltage, DIBL with the channel width, height, doping concentration considering and not considering quantum confinement effect are studied, respectively. The results show influences of quantum confinement effect on source/drain depletion regions, threshold voltage and DIBL. The developed model will offer quantum corrections in JL SG nanosheet NMOSFET. [ABSTRACT FROM AUTHOR]

[Display omitted] • New phase field model for understanding pitting corrosion phenomena is proposed. • Using temperature-dependent current density to include the effect of temperature. • Accuracy of the model is verified versus experiments and other numerical results. • Several pitting corrosion problems with complex pit morphology are presented. This study presents a new phase field model of pitting corrosion for metal materials, encapsulating the transport of ionic species and electrochemical reactions in the electrolyte, and the effect of temperature on corrosion kinetics is incorporated by relating the interface kinetics parameter to the temperature. The Allen-Cahn and Cahn-Hilliard equations, along with the Poisson's equation, is established to govern phase transformation, ions diffusion and electrostatic potential distribution, respectively. The model is validated by conducting several benchmark numerical studies. Simulation results demonstrate that within the temperature range of 10 °C to 20 °C, the model closely matches experimental data and result from sharp interface model. Additionally, the model effectively reproduces the pit shapes observed in experiments and captures variations in ion concentrations within the electrolyte. Several examples of the model simulating corrosion problems with complex evolving morphologies are provided. [ABSTRACT FROM AUTHOR]

A displacement perturbation method is proposed for phase-field simulations of the ferroelectric domain structures. A temperature-misfit strain phase diagram is computed to validate this method's accuracy, encompassing rhombohedral, orthorhombic, tetragonal, cubic, and mixed phases by comparing with previous phase-field simulations. The change in free energy density surface with temperature and strain distribution is computed to clarify the mechanism of the mixed phase. The phase diagram, ferroelectric hysteresis, and domain structure all demonstrate that the numerical method of displacement discretization is reliable and practical in solving the time-dependent Ginzburg–Landau equation, consistent with previous simulated and experimental results. This new method offers the advantages of simple programming and easy parallelism, paving the way for advancements in phase-field modeling. A new generalized displacement perturbation method has been proposed for phase-field modeling to describe the polarization transition and evolution of the ferroelectric domain structure under an external electric field. The phase diagram, ferroelectric hysteresis, and domain structure all indicate that the numerical method of displacement discretization is reliable and practical in solving the time-dependent Ginzburg–Landau equation, consistent with previous simulated and experimental results. The new method boasts the advantages of simple programming and easy parallelism, which could pave the way for the development of phase-field modeling. [Display omitted] [ABSTRACT FROM AUTHOR]

This paper presents a fourth-order compact immersed interface method to solve two-dimensional Poisson equations with discontinuous solutions on arbitrary domains divided by an interface. The compact scheme only employs a nine-point stencil for each grid point on the computational domain. The new approach is based on an implicit formulation obtained from generalized Taylor series expansions, and it is constructed from a few modifications to the central finite difference near the interface. The discretization results in a linear system in which the matrix coefficients are the same as the ones for smooth solutions, and the right-hand side system is modified by adding terms known as jump contributions. These contributions are only calculated at those points where the nine-point stencil cuts the interface. However, the contribution formulas require the knowledge of Cartesian jumps up to fourth-order. In this paper, we derived them using only the principal jump conditions and the jumps coming from the known right-hand function of the Poisson equation. We present numerical experiments in two dimensions to verify the feasibility and accuracy of the proposed method. Thus, the implicit immersed interface method results in an attractive fourth-order compact scheme that is easy to be implemented and applied to arbitrary interface shapes. [ABSTRACT FROM AUTHOR]

In this paper, both the standard finite element method (FEM) and the mixed FEM are developed for the fourth-order modified Poisson-Fermi equation resulted from the Bazant-Storey-Kornyshev (BSK) theory to account for electrostatic correlations in concentrated electrolytes. Optimal convergence properties are obtained for both FEMs in their respective norms, additionally, the mixed FEM can produce one-order higher approximation accuracy for the electric field in L 2 norm than that of the standard FEM, resulting in more accurate approximations to the electrostatic stress and the force of interaction in concentrated electrolytes. All attained theoretical results are validated by numerical experiments. Furthermore, a practical example is studied to validate the necessity of introducing the fourth-order modified Poisson-Fermi equation to describe the electrostatic potential field due to correlation effects by comparing with the classical Poisson equation that accounts for the classical mean-field Poisson-Boltzmann (PB) theory, illustrating that numerical trends obtained from the modified model are all consistent with experimental results under the circumstance of electrostatic correlations in concentrated electrolytes, which however cannot be predicted by the classical model. [ABSTRACT FROM AUTHOR]

An extrapolation accelerated multiscale Newton-multigrid (EMNMG) method is proposed to solve two-dimensional semilinear Poisson equations. The nine-point fourth-order compact schemes are used to approximate the nonlinear Poisson equations. In order to accelerate Newton-MG method for calculating the finite difference (FD) solution on the finest grid, a quite good initial guess is constructed from the fourth-order FD solutions at two coarse levels by using Richardson extrapolation and bi-quartic polynomial interpolation, which greatly reduces the number of Newton iterations required. A completed extrapolation technique is adopted to generate a sixth-order extrapolated solution on entire finest grid cheaply. Numerical results are given to show that our proposed EMNMG algorithm can achieve high accuracy and keep less cost simultaneously. [ABSTRACT FROM AUTHOR]

In this paper we present the distribution of the maximum of the asymmetric telegraph process in an arbitrary time interval [ 0 , t ] under the conditions that the initial velocity V (0) is either c 1 or − c 2 and the number of changes of direction is odd or even. For the case V (0) = − c 2 the singular component of the distribution of the maximum displays an unexpected cyclic behavior and depends only on c 1 and c 2 , but not on the current time t. We obtain also the unconditional distribution of the maximum for either V (0) = c 1 or V (0) = − c 2 and its expression has the form of series of Bessel functions. We also show that all the conditional distributions emerging in this analysis are governed by generalized Euler–Poisson–Darboux equations. We recover all the distributions of the maximum of the symmetric telegraph process as particular cases of the present paper. We underline that it rarely happens to obtain explicitly the distribution of the maximum of a process. For this reason the results on the range of oscillations of a natural process like the telegraph model make it useful for many applications. [ABSTRACT FROM AUTHOR]

A shock wave can be formed by continuous mass loading of the solar wind by the newly formed cometary ions. The formation of ion acoustic shocklets has thus been studied in a plasma of solar and cometary electrons, described by kappa distribution functions with different temperatures and spectral indices, a drifting H 3 O + ion component and a pair of oppositely charged oxygen ion components. The Korteweg-deVries-Burger's equation which describes weakly nonlinear waves in a dissipative medium has been derived for the above plasma composition using the momentum, continuity and Poisson's equations and studied for parameters observed at the inner shock region of comet Halley. We find that the spectral index of the cometary electrons plays a key role in the formation, speed of propagation and width of the shock wave, which has been interpreted as a "shocklet", whose strength decreases as the spectral indices increase or as the suprathermal distribution relaxes to a Maxwellian distribution, with a buildup of colder electrons. Also, all three components of ions contribute to the formation of shocklets; thus bringing out the additive nature of the contributions of various types of ions to their formation. This study could aid the understanding of in-situ measurements of shock waves in cometary plasmas. [ABSTRACT FROM AUTHOR]

Poisson-Nernst-Planck equations are a nonlinear coupled system which are widely used to describe electrodiffusion processes in biomolecular systems and semiconductors, etc. A virtual element method with order k (k ≥ 1) is proposed to numerically approximate the Poisson-Nernst-Planck equations on polygonal meshes. The error estimates in the H 1 norm are presented for the numerical solution to the Poisson-Nernst-Planck equations. Numerical examples show that the virtual element method can work well on general polygonal elements and the numerical results agree with the theoretical prediction. [ABSTRACT FROM AUTHOR]

• Laminar, fully developed two-phase flow in an arbitrary cross-section tube. • Behavior of the velocity and shear stress investigated near the triple point. • Local treatment and solution obtained independent of the tube's cross section. • Mathematical tools used to obtain a simple analytical solution. In this study, we propose a simple approach for examining the local behavior of the velocity and shear stresses in the vicinity of a triple point (TP) in a two-phase flow. We assume a laminar steady, fully developed, stratified two-phase flow with any contact angle, α , independent of the shape of the tube's cross-section. The axial component of the fluid's velocity can be treated as a scalar function in two dimensions that obeys Poisson's equation, together with the no-slip condition on the boundary (tube walls) and appropriate conditions on the interface between the phases. In our opinion, the exact solutions described in previous studies (for the special case of a cylindrical tube) are complicated and difficult to follow. Moreover, the approximated interface presented in previous studies predicts a geometric contact angle (α g e o m e t r y) that might not coincide with the physical contact angle, α , which is located at the real interface's shape. The exact solutions numerically solve the global problem for a circular tube (i.e., they find the velocity and shear stresses) and analytically obtain results in the vicinity of the TP to solve the problem with the aid of the residue theorem. However, we begin in the vicinity of the TP and our entire analysis focuses within this domain. To simplify the calculations, three steps are suggested: 1) zooming in on the TP allows us to approximate the curved arcs of the walls and the interface as straight lines (the tangents of the arcs of the wall and the outward ray of the interface) 1 1 Intuitively, for a local observation up to first order in distance, a regular curve can be approximated as a straight line; i.e., an arc can be approximated to a cord. Up to second order, a curve can be approximated as a circle. For a formal proof that Laplace's equation for the region is bounded by the wall circle (radius R w a l l), the interface circle (radius R i), and our radius R defined in Fig 2.2, a possible option is to Möbius-transform the wall circle and the interface circle into straight lines concurrent in the angle α (so the other meeting points of these circles are mapped onto the point at infinity), and to obtain the Laplace solution, thereby proving that when R / (min (R w a l l , R i n t e r f a c e)) < < 1 , the deviation tends to zero. ; 2) attaching a polar coordinate system where the origin of this coordinate system is located at the TP; and 3) suppressing the local driving forces and expressing distant driving forces as a function on the boundary, thereby leaving the velocity as a harmonic function. The proposed method can be converted for electromagnetism near a junction of several dielectric materials. Furthermore, the electromagnetic parallel problem can be extended to three dimensions; i.e., a junction point of prisms with different dielectric constants. [ABSTRACT FROM AUTHOR]

• A meshfree method based on a non-boundary-fitted discretization is proposed. • The proposed method is efficient for simulation of complex geometries. • The micro-CT scan images of the cellular materials can be directly employed. • The solution is approximated by library Exponential Basis Functions (EBFs). • The approximation is enriched by the modified singular functions. A meshfree method on fixed grids is devised for simulation of Poisson's equation on 3D image-based cellular materials. The non-boundary fitted discretization of such jagged voxel models of complex geometries is accomplished through embedding the micro-CT scan image in a Cartesian grid of nodes. The computational nodes inside the solid voxels are found by a simple point-in-membership test. Using a set of modified singular functions around the voids, along with the library-rational-exponential basis functions (EBFs) satisfying the governing differential equation, an enriched spatial solution is locally constructed on a generic computational cloud/cell (GCC) of nodes containing the voids. Within each GCC, the boundary conditions are satisfied through a weighted least-squares approximation. Finally, by establishing point-wise compatibility between the solutions of the GCCs a well-conditioned small linear system of equations is resulted. The results are compared with those of the finite element method (FEM) using extremely fine meshes with an excessive number of nodes. [ABSTRACT FROM AUTHOR]

The multinode Shepard operator is a linear combination of local polynomial interpolants with inverse distance weighting basis functions. This operator can be rewritten as a blend of function values with cardinal basis functions, which are a combination of the inverse distance weighting basis functions with multivariate Lagrange fundamental polynomials. The key for simply computing the latter, on a unisolvent set of points, is to use a translation of the canonical polynomial basis and the P A = L U factorization of the associated Vandermonde matrix. In this paper, we propose a method to numerically solve a Poisson equation with Dirichlet conditions through multinode Shepard interpolants by collocation. This collocation method gives rise to a collocation matrix with many zero entrances and a smaller condition number with respect to the one of the well known Kansa method. Numerical experiments show the accuracy and the performance of the proposed collocation method. [ABSTRACT FROM AUTHOR]