Self-adjoint Elliptic Equations Research Articles

Higher-order adaptive virtual element methods with contraction properties

<abstract><p>The realization of a standard Adaptive Finite Element Method (AFEM) preserves the mesh conformity by performing a completion step in the refinement loop: In addition to elements marked for refinement due to their contribution to the global error estimator, other elements are refined. In the new perspective opened by the introduction of Virtual Element Methods (VEM), elements with hanging nodes can be viewed as polygons with aligned edges, carrying virtual functions together with standard polynomial functions. The potential advantage is that all activated degrees of freedom are motivated by error reduction, not just by geometric reasons. This point of view is at the basis of the paper [L. Beirão da Veiga et al., "Adaptive VEM: stabilization-free a posteriori error analysis and contraction property", SIAM Journal on Numerical Analysis, vol. 61, 2023], devoted to the convergence analysis of an adaptive VEM generated by the successive newest-vertex bisections of triangular elements without applying completion, in the lowest-order case (polynomial degree $ k = 1 $). The purpose of this paper is to extend these results to the case of VEMs of order $ k\ge2 $ built on triangular meshes. The problem at hand is a variable-coefficient, second-order self-adjoint elliptic equation with Dirichlet boundary conditions; the data of the problem are assumed to be piecewise polynomials of degree $ k-1 $. By extending the concept of global index of a hanging node, under an admissibility assumption of the mesh, we derive a stabilization-free a posteriori error estimator. This is the sum of residual-type terms and certain virtual inconsistency terms (which vanish for $ k = 1 $). We define an adaptive VEM of order $ k $ based on this estimator, and we prove its convergence by establishing a contraction result for a linear combination of (squared) energy norm of the error, (squared) residual estimator, and (squared) virtual inconsistency estimator.</p></abstract>

A class of finite element methods with averaging techniques for solving the three-dimensional drift-diffusion model in semiconductor device simulations

Obtaining a satisfactory numerical solution of the classical three-dimensional drift-diffusion (DD) model, widely used in semiconductor device simulations, is still challenging nowadays, especially when the convection dominates the diffusion. In this work, we propose a series of finite element schemes with different types of averaging techniques to discretize the three-dimensional continuity equations. Our methods are based on the classical finite element framework, quite different from those mixed finite element/volume methods that also employ inverse averaging techniques. At first, the Slotboom variables are employed to transform the continuity equations into self-adjoint second-order elliptic equations with exponentially behaved coefficients. Then four averaging techniques, denoted with A1-A4, are introduced to approximate the exponential coefficient with its average on every tetrahedral element of the grid. The first scheme calculates the harmonic average of the exponential coefficient on a whole tetrahedron, and the other three schemes calculate the average of the exponential coefficient on each edge of a tetrahedral element. Our methods can avoid the spurious non-physical numerical oscillations and guarantee the conservation of the computed terminal currents with a terminal current evaluation approach. In fact, these methods not only maintain numerical stability but also overcome the disadvantages of some stabilization methods that cannot guarantee the conservation of the terminal currents, such as the streamline-upwind Petrov-Galerkin (SUPG) method. Moreover, the derivation of these discretization methods does not need the dual Voronoi grid as that of the finite volume Scharfetter-Gummel (FVSG) method or other mixed finite element/volume methods with inverse averaging techniques, which greatly reduces the complexity of parallel implementations of our methods. Simulations on two realistic three-dimensional semiconductor devices are carried out to test the accuracy and stability of our methods. According to numerical results, we conclude that scheme A4 can produce more accurate numerical solutions than the other three schemes, especially when the bias applied on the electrode is high. Numerical results also show that the scheme A4 is more robust than the Zlámal finite element method [1] in high-bias cases, and it also performs better than the FVSG method and a tetrahedral mixed finite element method [2] on poor-quality grids. Scheme A4 is also employed to study rich physical properties of the n-channel MOSFET.

An inverse averaging finite element method for solving three-dimensional Poisson-Nernst-Planck equations in nanopore system simulations.

The Poisson-Nernst-Planck (PNP) model plays an important role in simulating nanopore systems. In nanopore simulations, the large-size nanopore system and convection-domination Nernst-Planck (NP) equations will bring convergence difficulties and numerical instability problems. Therefore, we propose an improved finite element method (FEM) with an inverse averaging technique to solve the three-dimensional PNP model, named inverse averaging FEM (IAFEM). At first, the Slotboom variables are introduced aiming at transforming non-symmetric NP equations into self-adjointsecond-order elliptic equations with exponentially behaved coefficients. Then, these exponential coefficients are approximated with their harmonic averages, which are calculated with an inverse averaging technique on every edge of each tetrahedral element in the grid. Our scheme shows good convergence when simulating single or porous nanopore systems. In addition, it is still stable when the NP equations are convection domination. Our method can also guarantee the conservation of computed currents well, which is the advantage that many stabilization schemes do not possess. Our numerical experiments on benchmark problems verify the accuracy and robustness of our scheme. The numerical results also show that the method performs better than the standard FEM when dealing with convection-domination problems. A successful simulation combined with realistic chemical experiments is also presented to illustrate that the IAFEM is still effective for three-dimensional interconnected nanopore systems.

Алгоритмы для решения параметрической самосопряжённой эллиптической краевой задачи в двумерной области методом конечных элементов высокого порядка точности

We consider the calculation schemes for solving elliptic boundary-value problems (BVPs) within the framework of the Kantorovich method that provides the reduction of an elliptic BVP to a system of coupled second-order ordinary differential equations (ODEs). The surface basis functions of the expansion depend on the independent variable of the ODEs parametrically. Here we use the basis functions calculated by means of the finite element method(FEM), as well as the probe parametric surface basis functions calculated in the analytical form. We propose new calculation schemes and algorithms for solving the parametric self-adjoint elliptic boundary-value problem (BVP) in a 2D finite domain, using high-accuracy finite element method (FEM) with rectangular and triangular elements. The algorithm and the programs calculate with the given accuracy the eigenvalues, the surface eigenfunctions and their first derivatives with respect to the parameter of the BVP for parametric self-adjoint elliptic differential equation with the Dirichlet and/or Neumann type boundary conditions on the 2D finite domain, and the potential matrix elements, expressed as integrals of the products of surface eigenfunctions and/or their first derivatives with respect to the parameter. The parametric eigenvalues (potential curves) and the potential matrix elements computed by the program can be used for solving bound-state and multi-channel scattering problems for systems of coupled second-order ODEs by means of the Kantorovich method. We demonstrate the efficiency of the proposed calculation schemes and algorithms in benchmark calculations of 2D elliptic BVPs describing quadrupole vibrations of a collective nuclear model.

Recovery type a posteriori estimates and superconvergence for nonconforming FEM of eigenvalue problems

The main goal of this paper is to present recovery type a posteriori error estimators and superconvergence for the nonconforming finite element eigenvalue approximation of self-adjoint elliptic equations by projection methods. Based on the superconvergence results of nonconforming finite element for the eigenfunction we derive superconvergence and recovery type a posteriori error estimates of the eigenvalue. The results are based on some regularity assumption for the elliptic problem and are applicable to the lowest order nonconforming finite element approximations of self-adjoint elliptic eigenvalue problems with quasi-regular partitions. Therefore, the results of this paper can be employed to provide useful a posteriori error estimators in practical computing under unstructured meshes.

Superconvergence of Solution Derivatives of the Shortley–Weller Difference Approximation to Elliptic Equations with Singularities Involving the Mixed Type of Boundary Conditions

This paper presents a superconvergence analysis for the Shortley–Weller finite difference approximation of second-order self-adjoint elliptic equations with unbounded derivatives on a polygonal domain with the mixed type of boundary conditions. In this analysis, we first formulate the method as a special finite element/volume method. We then analyze the convergence of the method in a finite element framework. An O(h 1.5)-order superconvergence of the solution derivatives in a discrete H 1 norm is obtained. Finally, numerical experiments are provided to support the theoretical convergence rate obtained.

On the Convergence of Finite Element Method for Second Order Elliptic Interface Problems

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L 2 and H 1 norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established.

A finite difference method for self-adjoint elliptic equations in three dimensions

In this article, we present a fourth-order finite difference method for the three-dimensional, second-order, self-adjoint elliptic partial differential equations subject to Dirichlet boundary conditions. A 19-point uniform cubic grid of size h is used to derive the method. The resulting system of algebraic equations could be solved by iterative methods. Numerical results of some test problems are given.

Single cell discretizations of order two and four for self-adjoint elliptic equations

Finite-difference schemes for the self-adjoint elliptic partial differential equation in two dimensions are developed. The methods use either five or nine grid points of a single square cell of size 2 h and have a truncation error of order h 2 or h 4. The resulting system of equations can be solved by iterative methods. Numerical results of two test problems with known analytical solutions are appended.

On using the cell discretization algorithm for mixed-boundary value problems and domain decomposition

The cell discretization algorithm is used to approximate solutions to self-adjoint elliptic equations with general nonhomogeneous Dirichlet or Neumann or mixed-boundary values. Error estimates are obtained showing general convergence. This provides the framework for a nonoverlapping iterative algorithm for domain decomposition. The domain of a Dirichlet problem is partitioned into two subdomains. Solutions on the two subdomains can be patched together to form a solution to the original problem provided the solutions agree across the common interface of the subdomains and the normal derivatives there have equal absolute values and are opposite in sign. We generate such a solution by alternating between imposing Neumann and Dirichlet conditions on the interface, with boundary data adapted from the results of the previous approximation. A posteriori error estimates show that we have global convergence provided computed interface errors are sufficiently small.

Finite Element Discretizations of Elliptic Problems in the Presence of Arbitrarily Small Ellipticity: An Error Analysis

The purpose of this paper is to analyze the error of the finite element method applied to the pressure equation arising in reservoir simulation. We study self-adjoint second-order elliptic equations with discontinuous coefficients and arbitrarily small (but uniformly positive) ellipticity. Under proper conditions on the permeability functions and the source term, we prove error estimates that are independent of the lower bound $\delta$ of the materiel coefficients. These results are based on an extensive regularity analysis of the interface problems of concern. More precisely, we show that the solution of our model problem is piecewise smooth and that the associated Sobolev norms are bounded independently of $\delta$. Finally, the error analysis is illustrated by numerical experiments.

DISCRETE CONSERVATION AND DISCRETE MAXIMUM PRINCIPLE FOR ELLIPTIC PDEs

A class of finite volume numerical schemes for the solution of self-adjoint elliptic equations is described. The main feature of the schemes is that numerical solutions share both discrete conservation and discrete strong maximum principle without restriction on the differential operator or on the volume elements.

Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization

The spectral element method is used to discretize self-adjoint elliptic equations in three-dimensional domains. The domain is decomposed into hexahedral elements, and in each of the elements the discretization space is the set of polynomials of degree N in each variable. A conforming Galerkin formulation is used, the corresponding integrals are computed approximately with Gauss--Lobatto--Legendre (GLL) quadrature rules of order N, and a Lagrange interpolation basis associated with the GLL nodes is used. Fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method. The conforming finite element space on the GLL mesh, consisting of piecewise Q1 or P1 functions, produces a stiffness matrix Kh that is known to be spectrally equivalent to the spectral element stiffness matrix KN. Kh is replaced by a preconditioner $\tilde{K}_h$ which is well adapted to parallel computer architectures. The preconditioned operator is then $\tilde{K}_h^{-1} K_N$. Techniques for nonregular meshes are developed, which make it possible to estimate the condition number of $\tilde{K}_h^{-1} K_N$, where $\tilde{K}_h$ is a standard finite element preconditioner of Kh , based on the GLL mesh. Two finite element--based preconditioners: the wirebasket method of Smith and the overlapping Schwarz algorithm for the spectral element method are given as examples of the use of these tools. Numerical experiments performed by Pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions.

The Block Stride Reduction (B.S.R.) AlgorithmWith Application To Domain DecompositionMethods

In this paper we consider the application of the Block Stride Reduction algorithm for the solution of separable self-adjoint elliptic equations on rectangular domains. Non-overlapping domain decomposition techniques are applied to reduce the differential operator to block diagonally bordered linear systems which can then be solved by the Preconditioned Conjugate Gradient method. Finally, a new preconditioner for such systems is outlined.

Numerical solution of non-separable elliptic equations by the iterative application of FFT methods

A method for the numerical solution of non-separable (self-adjoint) elliptic equations is described in which the basic approach is the iterative application of direct methods. Such equations may be transformed into Helmholtz form and this Helmholtz problem is solved by the iterative application of FFT methods. An equation which is ‘near’ (in some sense) to the Helmholtz equation is appropriately chosen from the general class of equations soluble directly by FFT methods (see, for example, Le Bail, 1972) and the iteration (of block-Jacobi form) consists of corrections to the relevant Fourier harmonic amplitudes of the solution of this ‘nearby’ equation. It is also shown that the method is equivalent to a D' Yakonov-Gunn iteration [D' Yakonov (1961), Gunn(1964)] with a particular choice of iteration parameter and it is well known that, for self-adjoint problems with smooth coefficients, this form of iteration has a convergence rate which is essentially independent of grid-size. In the Concus and Golub (1973)...

A finite volume convergence theory for the fast adaptive composite grid methods

The fast adaptive composite grid method (FAC) is an efficient multilevel technique for adaptive solution of partial differential equations. Most existing FAC convergence theory assumes that the discrete equations and interlevel transfers are variational, which restricts applicability primarily to Galerkin finite element discretizations of self-adjoint elliptic equations. This paper extends the variational two-level theory to the case that the so-called finite volume element scheme is used for discretization.

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation, Au=f , in a smooth region Ω < n ( n ...

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth regionΩ?<? n (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that isΩ h , an approximation of Ω with dist (Ω,Ω h )?Ch 2 is not in general a union of elements; and assumingu?H 4 (Ω) we show that one can recover the total flux across a segment of the boundary of Ω with an error ofO(h 2). We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.

Composite iterative method for elliptic problems in irregular regions

A finite element piecewise linear approximation to a self-adjoint second-order elliptic equation in an irregular bounded two-dimensional region with Dirichlet boundary condition is considered. A composite inner-outer iterations method for the arising linear systems of algebraic equations is presented. Preconditioned Richardson iterations are chosen as the outer iterations, in which the discrete Laplacian is employed as a preconditioner. In each outer iteration, systems with the Laplacian in an irregular region are solved by the imbedding capacitance matrix method that uses conjugate gradients as the inner iterations. We prove that with an optimal control of the termination criterium for inner iterations the rate of convergence of this composite iterative process depends only logarithmically on h, the parameter of triangulation. Extensive numerical experiments illustrate the theoretical predictions.

An incomplete factorization iterative technique for elliptic equations

This paper describes efficient iterative techniques for solving the large sparse symmetric linear systems that arise from application of finite difference approximations to self-adjoint elliptic equations. We use an incomplete factorization technique with the method of D'Yakonov type, generalized conjugate gradient and Chebyshev semi-iterative methods. We compare these methods with numerical examples. Bounds for the 4-norm of the error vector of the Chebyshev semi-iterative method in terms of the spectral radius of the iteration matrix are derived.