Two-dimensional Elliptic Problems Research Articles

Sparsifiable spectral equivalence of DtN mapping and its application to elliptic problems

A sparsifiable spectrally equivalent preconditioner is developed for a special finite element discretization of two-dimensional elliptic problems. We start with a spectrally equivalent preconditioner which is closely related to the discrete Dirichlet-to-Neumann (DtN) mappings in sub-domains. The fullness of discrete DtN mappings renders an essential difficulty for a direct implementation. Based on numerical evidences, we present a particular square root matrix that is spectrally equivalent to the discrete DtN mappings. Unfortunately, this square root matrix is still not sparsifiable. To remedy this problem, we resort to the Chebyshev rational approximation. Incorporated with the fast sine transforms, a sparsifiable and spectrally equivalent preconditioner is finally obtained. Numerical examples are provided to validate efficiency of the proposed preconditioning technique.

Multinode Shepard method for two-dimensional elliptic boundary problems on different shaped domains

In this paper, we continue the study on the application of multinode Shepard method to numerically solve elliptic Partial Differential Equations (PDEs) equipped with various conditions at the boundary of domains of different shapes. In particular, for the first time, the multinode Shepard method is proposed to solve elliptic PDEs with Dirichlet and/or Neumann boundary conditions. The method has been opportunely handled to efficiently work dealing with scattered distribution of points and, to this aim, several experiments in different 2d domains have been performed. Comparisons with the analytic solution and the results generated by the Kansa’s RBF solvers have been reported referring to Halton points. The results are very promising and should be of interest for applications in the real world.

The random feature method for solving interface problems

Interface problems have long been a major focus of scientific computing, leading to the development of various numerical methods. Traditional mesh-based methods often employ time-consuming body-fitted meshes with standard discretization schemes or unfitted meshes with tailored schemes to achieve controllable accuracy and convergence rate. Along another line, mesh-free methods bypass mesh generation but lack robustness in terms of convergence and accuracy due to the low regularity of solutions. In this study, we propose a novel method for solving interface problems within the framework of the random feature method (RFM). This approach utilizes random feature functions in conjunction with a partition of unity as approximation functions, and solves a linear least-squares system to obtain the approximate solution. In the context of interface problems, two innovative and crucial components are incorporated into the RFM. Firstly, we utilize two sets of random feature functions on each side of the interface, allowing for the inclusion of low regularity or even discontinuous behaviors in the solution. Secondly, the construction of the loss function is based on the assessment of the partial differential equation, initial/boundary conditions, and the interface condition on collocation points. This approach ensures that these conditions are equally satisfied. Consequently, the challenges arising from geometric complexity primarily manifest in the generation of collocation points, a task amenable to standard methods. Importantly, the proposed method retains its meshfree characteristics and robustness when addressing problems featuring intricate geometries. We validate our method through a series of linear interface problems with increasingly complex geometries, including two-dimensional elliptic and three-dimensional Stokes interface problems, a three-dimensional elasticity interface problem, a moving interface problem with topological change, a dynamic interface problem with large deformation, and a linear fluid–solid interaction problem with complex geometry. Our findings show that despite the solution often being only continuous or even discontinuous, our method not only eliminates the need for mesh generation but also maintains high accuracy, akin to the spectral collocation method for smooth solutions. Remarkably, for the same accuracy requirement, our method requires two to three orders of magnitude fewer degrees of freedom than traditional methods, demonstrating its significant potential for solving interface problems with complex geometries and predetermined intricate evolution.

FDM-PINN: Physics-informed neural network based on fictitious domain method

In this article, we present a physics-informed neural network combined with fictitious domain method (FDM-PINN) to study linear elliptic and parabolic problems with Robin boundary condition. Our goal here to develop a deep learning framework where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction on small domain , which is geometrically simple shaped domain. We study the applicability and accuracy of FDM-PINN for the elliptic and parabolic problems with fixed ω or moving ω. This method is of the virtual control type and relies on a well-designed neural network. Numerical results obtained by FDM-PINN for two-dimensional elliptic and parabolic problems are given, which are more accurate than the results obtained by least-squares/fictitious domain method in [R. Glowinski and Q. He, A least-squares/fictitious domain method for linear elliptic problems with robin boundary conditions, Commun. Comput. Phys. 9 (2011), pp. 587–606.].

An Efficient Red–Black Skewed Modified Accelerated Arithmetic Mean Iterative Method for Solving Two-Dimensional Poisson Equation

This paper presents the extended variants to the established two-stage Arithmetic Mean (AM) method known as the Modified Accelerated Arithmetic Mean (MAAM) and Skewed Modified Accelerated Arithmetic Mean (SkMAAM) methods to solve the two-dimensional elliptic problem. The existing two-stage AM and its skewed variants apply one weighted parameter for the computation of nodes in Levels 1 and 2. The suggested MAAM and SkMAAM methods employ red–black ordering with two different weighted parameters and an additional two distinct accelerated parameters for red and black nodes, respectively. By carefully choosing optimum parameter values, the proposed MAAM and SkMAAM improve the computational execution of the algorithm. With red–black ordering, the computational molecules of red and black nodes are symmetrical, in which the computation of red nodes applies the updated values of their four neighbouring black nodes and vice versa. These symmetrical computational molecules of red and black nodes can be seen for the modified variants MAM and MAAM, and their corresponding skewed variants SkMAM and SkMAAM. The proposed MAAM and SkMAAM methods are compared to the existing AM and Modified AM (MAM) and their corresponding skewed variants, namely the Skewed AM (SkAM) and Skewed MAM (SkMAM) methods. The performance of the newly proposed MAAM and SkMAAM methods is compared against the existing methods in terms of computational complexity and actual execution time. It is shown in the simulation results that the skewed variants are superior to their corresponding regular variants, in which the SkMAAM method gives the best performance.

Adaptive multilayer neural network for solving elliptic partial differential equations with different boundary conditions

In recent years, elliptic partial differential equations (PDEs) have been challenging in numerical mathematics, engineering, and physics. In this paper, we proposed a novel approach called Multilayer Neural Network for Partial Differential Equations (MLNPDE) based on deep learning algorithms by exploring the elliptic PDE family under certain conditions such as initial conditions and boundary conditions to govern approximation solutions of PDEs. In the proposed model, we include a multilayer neural network using a densely connected network. Moreover, the approximation solution underlying PDEs can be expressed in two terms: the first term satisfies the boundary conditions, and the second term is a function of the unknown parameters that were estimated by our proposed model. Furthermore, we derive a normal form of the boundary conditions on the first term of the approximation solution for two-dimensional elliptic partial differential equation problems with an arbitrary domain. The numerical results show that our method achieved tremendous state-of-the-art performance in accuracy and efficiency compared to the classical methods.

A novel higher order compact-immersed interface approach for elliptic problems

We present a new higher-order accurate finite difference explicit jump Immersed Interface Method for solving two-dimensional elliptic problems with singular source and discontinuous coefficients in the irregular region on a compact Cartesian mesh. We propose a new strategy for discretizing the solution at irregular points on a nine-point compact stencil such that the higher-order compactness is maintained throughout the whole computational domain. The scheme is employed to solve four problems embedded with circular- and star-shaped interfaces in a rectangular region having analytical solutions and varied discontinuities across the interface in source and the coefficient terms. We also simulate a plethora of fluid flow problems past bluff bodies in complex flow situations which are governed by the Navier–Stokes equations; they include problems involving multiple bodies immersed in the flow as well. In the process, we show the superiority of the proposed strategy over the explicit jump finite difference immersed interface method and other existing immersed interface methods by establishing the rate of convergence and grid independence of the computed solutions. In all the cases, our computed results are extremely close to the available numerical and experimental results.

High Order Compact Block-Centered Finite Difference Schemes for Elliptic and Parabolic Problems

Based on the combination of block-centered and compact difference methods, fourth order compact block-centered finite difference schemes for the numerical solutions of one-dimensional and two-dimensional elliptic and parabolic problems with variable coefficients are derived and analyzed. Stability and optimal fourth-order error estimates are proved for both the solution and flux. Numerical experiments for model problems are presented to confirm the theoretical results and superior performance of the proposed schemes.

Anisotropic a posteriori error estimate for the virtual element method

Abstract We derive an anisotropic a posteriori error estimate for the adaptive conforming virtual element approximation of a paradigmatic two-dimensional elliptic problem. In particular, we introduce a quasi-interpolant operator and exploit its approximation results to prove the reliability of the error indicator. We design and implement the corresponding adaptive polygonal anisotropic algorithm. Several numerical tests assess the superiority of the proposed algorithm in comparison with standard polygonal isotropic mesh refinement schemes.

A stable RBF partition of unity local method for elliptic interface problems in two dimensions

The interface problems are faced with multiple connected domains and consequently, their solutions or derivatives might be discontinuous. This paper proposes the use of collocation based radial basis function partition of unity method (RBF-PUM) for solving two-dimensional elliptic interface problems. The RBF-PUM is a local method that allows overcoming the high computational cost associated with the global RBF methods. In the RBF-PUM, the domain is split into overlapping patches forming a covering of it. However, this method suffers from instability when the RBF shape parameter ε tends to zero. To overcome this issue, we use the RBF-QR algorithm which offers stable computations for all values of ε and provides higher accuracy. To obtain the appropriate solution in the vicinity of the interface, the domain decomposition technique is used. In this technique, the approximation in each subdomain is built separately, and proper jump conditions are then imposed across the interface. We illustrate how to apply the proposed method to Sturm-Liouville, Sturm-Liouville eigenvalue and elastostatic interface problems. The proposed method in dealing with arbitrary interfaces within different domain sizes is validated. We present some numerical examples in which the results are compared with exact solutions and those provided by other numerical methods.

A global RBF-QR collocation technique for solving two-dimensional elliptic problems involving arbitrary interface

In recent years, several numerical methods have been developed for differential equations with discontinuous coefficients, which are known as interface problems. Our aim is to introduce an efficient global meshless method for solving elliptic problems with arbitrary interface. Meshless methods based on infinitely smooth radial basis functions (RBFs) can achieve spectral convergence rate for solving partial differential equations. RBF-QR is a newly developed method which allows numerically stable computations with RBFs for all values of the free shape parameter $$\varepsilon$$. In this paper, an appropriate RBF-QR collocation technique is applied for two-dimensional elliptic interface problems, including Poisson, Helmholtz and elasticity problems. The Helmholtz interface problem is studied, by taking into account the fact that the solutions of such problems might be oscillatory. The elasticity interface problem is discussed as an application of elliptic interface problem in vector form. Several problems involving complex interface geometries and discontinuous variable coefficients are presented to confirm the efficiency and stability of RBF-QR method for this class of problems in the case of arbitrary small values of $$\varepsilon$$.

Mid-range natural orbits around the triple asteroid 2001SN263

The goal of the present paper is to search for natural trajectories that can be used to place a spacecraft to observe the triple asteroid 2001SN263. For a first set of observations mid-range distance orbits are ideal. To model this triple asteroid from a mid-range distance, it is considered that it can be modeled as a single ellipsoid body, which gives a simple mathematical model that allows on-board computation of trajectories. The mass is assumed to be equal to the sum of the masses of the three bodies and its dimensions are calculated considering the size of the orbits of Beta and Alpha. The objective is to search for orbits that can be used at the arrival time of the spacecraft in the vicinity of the triple asteroid to observe the bodies to make a better evaluation of their dimensions and dynamics before a good choice for a final orbit can be made. The two-dimensional elliptic restricted three-body problem is used as the mathematical model with the addition of the perturbations coming from the asteroid and the Sun irregular gravity fields (J2). It is also considered the disturbance caused by the solar radiation pressure, a major force near asteroids.

TWO-DIMENSIONAL SPLITTING SCHEMES FOR HYPERBOLIC EQUATIONS

The article considers splitting schemes in geometric directions that approximate the initial-boundary value problem for p-dimensional hyperbolic equation by chain of two-dimensional-one-dimensional problems. Two ways of constructing splitting schemes are considered with an operator factorized on the upper layer, algebraically equivalent to the alternating direction scheme, and additive schemes of total approximation. For the first scheme, the restrictions on the shape of the region G at p=3 can be weakened in comparison with schemes of alternating directions, which are a chain of three-point problems on the upper time layer, the region can be a connected union of cylindrical regions with generators parallel to the axis OX3. In the second case, for a three-dimensional equation of hyperbolic type, an additive scheme is constructed, which is a chain «two-dimensional problem – one-dimensional problem» and approximates the original problem in a summary sense (at integer time steps). For the numerical implementation of the constructed schemes – the numerical solution of two-dimensional elliptic problems – one can use fast direct methods based on the Fourier algorithm, cyclic reduction methods for three-point vector equations, combinations of these methods, and other methods. The proposed two-dimensional splitting schemes in a number of cases turn out to be more economical in terms of total time expenditures, including the time for performing computations and exchanges of information between processors, compared to traditional splitting schemes based on the use of three-point difference problems for multiprocessor computing systems, with different structures of connections between processors type «ruler», «matrix», «cube», with universal switching.

A Meshfree Computational Approach Based on Multiple-Scale Pascal Polynomials for Numerical Solution of a 2D Elliptic Problem with Nonlocal Boundary Conditions

A two-dimensional (2D) elliptic problem with nonlocal boundary conditions on both regular and irregular domains is solved numerically by Pascal polynomial basis unified with multiple-scale technique. Very accurate numerical solutions and quite reasonable condition numbers are obtained with the proposed method which is also a truly meshfree method since difficult meshing processes or numerical integrations over domains are not needed for considered problems. Four test problems are solved to show the accuracy and efficiency of the proposed method. Also stability of the method is studied against large noise effect.

An augmented matched interface and boundary (MIB) method for solving elliptic interface problem

Abstract In this paper, a second order accurate augmented matched interface and boundary (MIB) is introduced for solving two-dimensional (2D) elliptic interface problems with piecewise constant coefficients. The augmented MIB seamlessly combines several key ingredients of the standard MIB, augmented immersed interface method (IIM), and explicit jump IIM, to produce a new fast interface algorithm. Based on the MIB, zeroth and first order jump conditions are enforced across an arbitrarily curved interface, which yields fictitious values on Cartesian nodes near the interface. By using such fictitious values, a simple procedure is proposed to reconstruct Cartesian derivative jumps as auxiliary variables and couple them with the jump-corrected Taylor series expansions, which allow us to restore the order of the central difference across the interface to two. Moreover, by using the Schur complement to disassociate the algebraic computation of auxiliary variables and function values, the discrete Laplacian can be efficiently inverted by using the fast Fourier transform (FFT). It is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size. As a consequence, the total computational cost of the augmented MIB is about O ( n 2 log n ) for a Cartesian grid with dimension n × n in 2D. Therefore, the augmented MIB outperforms the classical MIB in all cases by significantly reducing the CPU time, while keeping the same second order of accuracy in dealing with complicated interfaces.

A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method

This paper is devoted to the development of an innovative Matlab software, dedicated to the numerical analysis of two-dimensional elliptic problems, by means of the probabilistic approach. This approach combines features of the Monte Carlo random walk method with discretization and approximation techniques, typical for meshless methods. It allows for determination of an approximate solution of elliptic equations at the specified point (or group of points), without a necessity to generate large system of equations for the entire problem domain. While the procedure is simple and fast, the final solution may suffer from both stochastic and discretization errors. The attached Matlab software is based on several original author’s concepts. It permits the use of arbitrarily irregular clouds of nodes, non-homogeneous right-hand side functions, mixed type of boundary conditions as well as variable material coefficients (of anisotropic materials). The paper is illustrated with results of analysis of selected elliptic problems, obtained by means of this software.

Vortex structures for some geometric flows from pseudo-Euclidean spaces

For some geometric flows (such as wave map equations, Schrödinger flows) from pseudo-Euclidean spaces to a unit sphere contained in a three-dimensional Euclidean space, we construct solutions with various vortex structures (vortex pairs, elliptic/hyperbolic vortex circles, and also elliptic vortex helices). The approaches base on the transformations associated with the symmetries of the nonlinear problems, which will lead to two-dimensional elliptic problems with resolution theory given by the finite-dimensional Lyapunov-Schmidt reduction method in nonlinear analysis.

Anisotropic Mesh Refinement Considering a Recovery-Based Error Estimator and Metric Tensors

This paper proposes a new anisotropic h-adaptive technique applied to the finite element method considering triangular finite elements. The proposed technique, named anisotropic error density recovery, utilizes the concept of the energy error density function and the analytical solution of an optimization problem by Lagrange’s multipliers to obtain a metric tensor that represents the dimensions of the new elements. This process aims to limit and equidistribute discretization errors throughout the mesh utilizing finite elements with large aspect ratios. The anisotropic h-adaptive technique is computationally implemented and tested in two-dimensional elliptic problems. The results demonstrate the ability of this approach to generate meshes of elements with high aspect ratios and oriented according to the principal directions of discretization errors. Thus, when compared with isotropic refinement techniques, the anisotropic error density recovery approach shows a significant reduction in the number of vertices for the same permissible error value.

Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

Abstract In this article we propose a novel nonstationary iterated Tikhonov (NIT)-type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: (I) a two-dimensional elliptic parameter identification problem (inverse potential problem); and (II) an image-deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

On a Family of Gradient-Type Projection Methods for Nonlinear Ill-Posed Problems

We propose and analyze a family of successive projection methods whose step direction is the same as the Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family encompasses the Landweber method, the minimal error method, and the steepest descent method; thus, providing an unified framework for the analysis of these methods. Moreover, we define new methods in this family, which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to successively project onto these sets. Numerical experiments are presented for a nonlinear two-dimensional elliptic parameter identification problem, validating the efficiency of our method.