Three-dimensional Elliptic Problems Research Articles

A meshless method based on the generalized finite difference method for three-dimensional elliptic interface problems

This article presents a meshless method to solve three-dimensional elliptic interface problem. The method is based on the generalized finite difference method, which expresses the derivatives of unknown variables by linear combinations of nearby function values. The proposed method turns the interface problem into some boundary value sub-problems coupled by the interface conditions. This conversion leads to an important feature, that is, the proposed method is not sensitive to the jump coefficients or the interface's geometry. It can handle different complex interfaces by only changing the level set function of the interface in the process. Several numerical examples with sufficient complexity verify the accuracy and stability of the method. For some given examples, the method is more accurate than the classical immersed finite element method.

A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems

In this paper, a fourth order augmented matched interface and boundary (AMIB) method is proposed for solving a three-dimensional elliptic interface problem which involves a smooth material interface inside a cuboid domain. On the boundary of the cuboid domain, the fourth order AMIB method can handle different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations in fictitious value generation. Moreover, zero-padding solutions are introduced so that the fast Fourier transform (FFT) algorithm is still valid near the boundary. In dealing with the interior interface, a fourth order ray-casting matched interface and boundary (MIB) scheme is proposed, which enforces the jump conditions along the normal direction for calculating fictitious values. Comparing with the existing MIB scheme, the ray-casting scheme naturally bypasses the corner issue and becomes more robust in handling complex geometry. Based on fictitious values generated near interface and boundary, the fourth order central difference can be corrected at various irregular points including corner points, by introducing Cartesian derivative jumps as auxiliary variables. This gives rise to an enlarged linear system, which can be efficiently solved by the Schur complement procedure together with the FFT inversion of the discrete Laplacian. Extensive numerical experiments have been carried to test the proposed ray-casting AMIB method for numerical accuracy, efficiency, and robustness in corner treatment. The numerical results demonstrate that the ray-casting AMIB scheme not only maintains a fourth order of accuracy in treating various interfaces and boundaries for both solutions and solution gradients, but also attains an overall efficiency on the order of O(n3log⁡n) for a n×n×n uniform grid.

Parallelized discrete exterior calculus for three-dimensional elliptic problems

A formulation of elliptic boundary value problems is used to develop the first discrete exterior calculus (DEC) library for massively parallel computations with 3D domains. This can be used for steady-state analysis of any physical process driven by the gradient of a scalar quantity, e.g. temperature, concentration, pressure or electric potential, and is easily extendable to transient analysis. In addition to offering this library to the community, we demonstrate one important benefit from the DEC formulation: effortless introduction of strong heterogeneities and discontinuities. These are typical for real materials, but challenging for widely used domain discretization schemes, such as finite elements. Specifically, we demonstrate the efficiency of the method for calculating the evolution of thermal conductivity of a solid with a growing crack population. Future development of the library will deal with transient problems, and more importantly with processes driven by gradients of vector quantities. Program summaryProgram Title: ParaGEMSCPC Library link to program files:https://doi.org/10.17632/t8rpkc7xnx.1Developer's repository link:https://bitbucket.org/pieterboom/paragems/Licensing provisions: BSD 2-clauseProgramming language: Fortran 90External routines/libraries: BLAS/LAPACK, MPI, PETSc [1], hypre ParaSails [2,3]Nature of problem: Large scale simulation of 3D elliptic boundary value problems in discrete media with heterogeneous or discontinuous properties.Solution method: ParaGEMS is a library implementing discrete exterior calculus (DEC) for distributed memory architectures. It is distributed with mini-applications for solving 3D elliptic boundary value problems, which describe a number of physical laws (Fourier's law, Fick's law, Darcy's law and Ohm's law), in discrete media with heterogeneous or discontinuous properties.Additional comments including restrictions and unusual features: ParaGEMS requires meshes generated in the default format used by Triangle [4] and TetGen [5].

Solving three-dimensional interface problems with immersed finite elements: A-priori error analysis

Immersed finite element methods are designed to solve interface problems on interface-unfitted meshes. However, most of the study, especially analysis, is mainly limited to the two-dimension case. In this paper, we provide an a priori analysis for the trilinear immersed finite element method to solve three-dimensional elliptic interface problems on Cartesian grids consisting of cuboids. We establish the trace and inverse inequalities for trilinear IFE functions for interface elements with arbitrary interface-cutting configuration. Optimal a priori error estimates are rigorously proved in both energy and L2 norms, with the constant in the error bound independent of the interface location and its dependence on coefficient contrast explicitly specified. Numerical examples are provided not only to verify our theoretical results but also to demonstrate the applicability of this IFE method in tackling some real-world 3D interface models.

Robust Alternating Direction Implicit Solver in Quantized Tensor Formats for a Three-Dimensional Elliptic PDE

The aim of this paper is to propose a robust numerical solver, which is capable of efficiently solving a three-dimensional elliptic problem in a data-sparse quantized tensor format. In particular, ...

An efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solution of 2-D and 3-D second order elliptic interface problems

In this work, we propose an efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solutions of two-dimensional (2-D) and three-dimensional (3-D) elliptic interface problems which may have discontinuous coefficients and curved interfaces with or without sharp corners. The proposed method uses Pascal polynomials as basis functions and utilizes multiple-scale approach for stabilizing numerical solutions. It is a well known fact that using polynomial basis without any modifications to obtain numerical solutions of partial differential equations may be peculiar owing to ill conditioned resultant coefficient matrix which is formed after process of discretization. Hence, as a remedy to the highly ill-conditioned coefficient matrix we employ multiple-scale approach. The main idea behind the multiple-scale approach is to make norm of all columns of resultant coefficient matrix equal to each other. The proposed method is a truly strong-form meshfree method since we do not need any mesh or integration process in problem domain, these features makes the implementation of the method very simple in computer environment. The efficiency of the proposed method is tested by some test problems which may have smooth interface or interface with sharp corners. Stability of the proposed method is investigated numerically in the presence of noise effect. Further, to show accuracy of the proposed method we present some comparisons with available numerical methods in literature, such as direct meshless local Petrov-Galerkin method, matched interface and boundary method, spectral element method and some meshless methods based on radial basis functions. The obtained numerical results and their comparisons confirm applicability of the proposed method for 2-D and 3-D steady state elliptic interface problems.

A Stabilized Dual Mixed Hybrid Finite Element Method with Lagrange Multipliers for Three-Dimensional Elliptic Problems with Internal Interfaces

This work studies an elliptic boundary value problem with diffusive, advective and reactive terms, in a three-dimensional domain composed of two media separated by a selective interface. For the numerical approximation of the problem we propose a novel approach that combines, for the first time: (1) a dual mixed hybrid (DMH) finite element method (FEM) based on the lowest order Raviart–Thomas space (RT0); (2) a Three-Field formulation; and (3) a Streamline Upwind/Petrov–Galerkin (SUPG) stabilization method. After proving that the weak formulation of the proposed method and its numerical counterpart are both uniquely solvable and that the finite element scheme enjoys optimal convergence properties with respect to the discretization parameter, we present an efficient implementation based on static condensation, which reduces the method to a nonconforming finite element approach on a grid made by three-dimensional simplices. Extensive computational tests indicate that: (1) the theoretical convergence properties are verified; (2) the DMH-RT0 FEM is accurate and stable even in the presence of marked interface jump discontinuities in the solution and its associated normal flux; and (3) in the case of strongly dominating advective terms, the SUPG stabilization resolves accurately steep boundary and/or interior layers without introducing spurious unphysical oscillations or excessive smearing of the solution front.

BDDC and FETI-DP preconditioners with adaptive coarse spaces for three-dimensional elliptic problems with oscillatory and high contrast coefficients

BDDC and FETI-DP algorithms are developed for three-dimensional elliptic problems with adaptively enriched coarse components. It is known that these enriched components are necessary in the development of robust preconditioners. To form the adaptive coarse components, carefully designed generalized eigenvalue problems are introduced for each faces and edges, and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance λTOL. Upper bounds for condition numbers of the preconditioned systems are shown to be CλTOL, with the constant C depending only on the maximum number of edges and faces per subdomain, and the maximum number of subdomains sharing an edge. Numerical results are presented to test the robustness of the proposed approach.

A numerical method for solving three-dimensional elliptic interface problems with triple junction points

Elliptic interface problems with multi-domains have wide applications in engineering and science. However, it is challenging for most existing methods to solve three-dimensional elliptic interface problems with multi-domains due to local geometric complexity, especially for problems with matrix coefficient and sharp-edged interface. There are some recent work in two dimensions for multi-domains and in three dimensions for two domains. However, the extension to three dimensional multi-domain elliptic interface problems is non-trivial. In this paper, we present an efficient non-traditional finite element method with non-body-fitting grids for three-dimensional elliptic interface problems with multi-domains. Numerical experiments show that this method achieves close to second order accurate in the L ∞ norm for piecewise smooth solutions.

An improved non-traditional finite element formulation for solving three-dimensional elliptic interface problems

Solving elliptic equations with interfaces has wide applications in engineering and science. The real world problems are mostly in three dimensions, while an efficient and accurate solver is a challenge. Some existing methods that work well in two dimensions are too complicated to be generalized to three dimensions. Although traditional finite element method using body-fitted grid is well-established, the expensive cost of mesh generation is an issue. In this paper, an efficient non-traditional finite element method with non-body-fitted grids is proposed to solve elliptic interface problems. The special cases when the interface cuts though grid points are handled carefully, rather than perturbing the cutting point away to apply the method for general case. Both Dirichlet and Neumann boundary conditions are considered. Numerical experiments show that this method is approximately second order accurate in the L∞ norm and L2 norm for piecewise smooth solutions. The large sparse matrix for our linear system also has nice structure and properties.

A Trilinear Immersed Finite Element Method for Solving the Electroencephalography Forward Problem

This article describes a trilinear immersed finite element method for solving the electroencephalography forward problem, which is a three-dimensional elliptic interface problem in the head geometry. The method uses hexahedral Cartesian meshes independent of the interfaces between head tissues, thus avoiding the sometimes difficult task of generating geometry fitting meshes. The interfaces are represented with level sets, and the finite element space is locally modified to better approximate the discontinuities of the solution. Numerical results show that this method achieves the same accuracy as the standard linear finite element method with geometry fitting mesh.

Multilevel algorithms for Rannacher–Turek finite element approximation of 3D elliptic problems

Generalizing the approach of a previous work of the authors, dealing with two-dimensional (2D) problems, we present multilevel preconditioners for three-dimensional (3D) elliptic problems discretized by a family of Rannacher Turek non-conforming finite elements. Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) lead to iterative methods which often have an optimal order computational complexity with respect to the number of degrees of freedom of the system. Such methods were first presented by Axelsson and Vassilevski in the late-1980s, and are based on (recursive) two-level splittings of the finite element space. An important point to make is that in the case of non-conforming elements the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general. To handle this, a proper two-level basis is required to enable us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method to the multilevel case. In the present paper new estimates of the constant γ in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality are derived that allow an efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver, also for the case of non-smooth coefficients. The second important achievement concerns the experimental study of AMLI solvers applied to the case of micro finite element (μFEM) simulation. Here the coefficient jumps are resolved on the finest mesh only and therefore the classical CBS inequality based convergence theory is not directly applicable. The obtained results, however, demonstrate the efficiency of the proposed algorithms in this case also, as is illustrated by an example of microstructure analysis of bones.

Matrix Decomposition MFS Algorithms

Matrix Decomposition MFS Algorithms

A New Block Parallel SOR Method and Its Analysis

As the development of the PSOR method (a point parallel SOR method by mesh domain partitioning proposed in [SIAM J. Sci. Comput., 20 (1999), pp. 2261-2281], this paper introduces a new mesh domain partition and ordering (the multitype partition and ordering), and proposes a new block parallel SOR (BPSOR) method for numerically solving 2-dimensional (2D) or three-dimensional (3D) elliptic boundary problems. A general mathematical analysis shows that the BPSOR method can have the same asymptotic convergence rate as the corresponding sequential block SOR method if the coefficient matrix of the block linear system is consistently ordered. It also shows that the original sequential ordering can be maintained in the parallel implementation of the BPSOR method so that the BPSOR method can be effectively applied to solve complex elliptic boundary problems. Furthermore, three particular multitype orderings are proposed based on strip and block mesh partitions, which lead to three effective BPSOR methods for solving the five-point like linear systems (in two dimensions) and the seven-point like linear systems (in three dimensions). In addition, it is shown that the PSOR method can be generated from BPSOR if each block equation is solved approximately by only one point SOR iteration. Thus, the PSOR method is well defined without involving any interprocessor data communication operations, and extended to solving three-dimensional (3D) problems. Finally, numerical results are presented which confirm the theoretical results and show that BPSOR has a good parallel performance on a parallel MIMD computer.

An adaptive multilevel wavelet collocation method for elliptic problems

An adaptive multilevel wavelet collocation method for solving multi-dimensional elliptic problems with localized structures is described. The method is based on multi-dimensional second generation wavelets, and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems [Int. J. Comp. Fluid Dyn. 17 (2003) 151]. Wavelet decomposition is used for grid adaptation and interpolation, while a hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid. In order to accelerate the convergence of the solver, an iterative procedure analogous to the multigrid algorithm is developed. The overall computational complexity of the solver is O ( N ) , where N is the number of adapted grid points. The accuracy and computational efficiency of the method are demonstrated for the solution of two- and three-dimensional elliptic test problems.

Efficient Solvers for Saddle-Point Problems Arising from Domain Decompositions with Lagrange Multipliers

In this paper, we are concerned with the domain decomposition method with Lagrange multipliers for solving three-dimensional elliptic problems with variable coefficients. We shall first introduce a weighted saddle-point problem resulting from this domain decomposition, which can be solved by existing iterative methods. Then we will construct two simple preconditioners, one for the system associated with the displacement variable and the other for the Schur complement system associated with the multiplier variable, that are applicable to various discretization schemes. The new preconditioners possess such local properties that they can be implemented cheaply. We will show that the condition number of the global preconditioned system grows only as the logarithm of the dimension of the local problem associated with an individual substructure and is independent of the large variations of the coefficients across the local interfaces.

Dual-Primal FETI Methods for Three-Dimensional Elliptic Problems with Heterogeneous Coefficients

In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dual-primal finite element tearing and interconnecting (FETI) methods which recently have been introduced and analyzed successfully for elliptic problems in the plane. The family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms. Particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual-primal preconditioner. It is shown that the condition numbers of several of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coefficients. These results closely parallel those of other successful iterative substructuring methods of primal as well as dual type.

Extension of the method of auxiliary mapping for three‐dimensional elliptic boundary value problems

In this paper, we extend the method of auxiliary mapping (MAM), introduced by Babuška and Oh, to three dimensions so that the extended MAM (3-D MAM) can effectively handle three-dimensional elliptic problems containing the singularities caused by the non-smooth domains. There are three type of singularities caused by non-smoothness of domains in R3: the vertex, the edge, and the vertex–edge combined singularities. To deal with the singularities of these types, we present three auxiliary mappings and formulas for the transformed bilinear forms and the transformed linear functionals by these auxiliary mappings. Then we present 3-D MAM and constructions of the blending-type elemental mappings for elements containing singularities. Numerical experiments that show the effectiveness of 3-D MAM are provided. Copyright © 2001 John Wiley & Sons, Ltd.

Extension of the method of auxiliary mapping for three-dimensional elliptic boundary value problems

SUMMARY In this paper, we extend the method of auxiliary mapping (MAM), introduced by Babuska and Oh, to three dimensions so that the extended MAM (3-D MAM) can eectively handle three-dimensional elliptic problems containing the singularities caused by the non-smooth domains. There are three type of singularities caused by non-smoothness of domains in R 3 : the vertex, the edge, and the vertex{edge combined singularities. To deal with the singularities of these types, we present three auxiliary mappings and formulas for the transformed bilinear forms and the transformed linear functionals by these auxiliary mappings. Then we present 3-D MAM and constructions of the blending-type elemental mappings for elements containing singularities. Numerical experiments that show the eectiveness of 3-D MAM are provided. Copyright ? 2001 John Wiley & Sons, Ltd.

Ordering Techniques for Two- and Three-Dimensional Convection-Dominated Elliptic Boundary Value Problems

Multigrid methods with simple smoothers have been proven to be very successful for elliptic problems with no or only moderate convection. In the presence of dominant convection or anisotropies as it might appear in equations of computational fluid dynamics (e.g. in the Navier-Stokes equations), the convergence rate typically decreases. This is due to a weakened smoothing property as well as to problems in the coarse grid correction. In order to obtain a multigrid method that is robust for convection-dominated problems, we construct efficient smoothers that obtain their favorable properties through an appropriate ordering of the unknowns. We propose several ordering techniques that work on the graph associated with the (convective part of the) stiffness matrix. The ordering algorithms provide a numbering together with a block structure which can be used for block iterative methods. We provide numerical results for the Stokes equations with a convective term illustrating the improved convergence properties of the multigrid algorithm when applied with an appropriate ordering of the unknowns.