Elliptic Differential Problems Research Articles

Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.

New approximations for one-dimensional 3-point and two-dimensional 5-point compact integrated RBF stencils

This paper presents some new compact approximation stencils based on integrated radial basis functions (IRBFs) for numerically solving second-order elliptic differential problems on Cartesian grids. Higher-order IRBF schemes are employed to approximate the field/dependent variable. The IRBF approximations in each direction are based on 3 points and constructed independently, where derivatives of the second, third, fourth, fifth and sixth orders along the grid line are enforced at the two end-points. The imposed nodal derivative values are simply acquired through a Picard-type iteration scheme. The stencil is made up of 3 points and 5 points for 1D and 2D discretisations, respectively. Numerical results show that the proposed stencils yield a high rate of convergence with respect to grid refinement, e.g. up to the 13th order for 1D problems and to the 9th order for 2D problems.

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T_n(f), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where mathbf {f} is an stimes s matrix-valued trigonometric polynomial with sge 1, and T_n(mathbf {f}) is the block Toeplitz matrix generated by mathbf {f}, whose size is N(n,s)=sn. The case s=1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the mathbb {Q}_p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

Spectral Analysis and Spectral Symbol of $d$-variate $\mathbb Q_{\boldsymbol p}$ Lagrangian FEM Stiffness Matrices

We study the spectral properties of the stiffness matrices coming from the approximation of a $d$-dimensional second order elliptic differential problem by the $\mathbb Q_{\boldsymbol p}$ Lagrangian finite element method (FEM); here, ${\boldsymbol p}=(p_1,\ldots,p_d)$ and $p_j$ represents the polynomial approximation degree in the $j$th direction. After presenting a construction of these matrices, we investigate the conditioning and the spectral distribution in the Weyl sense, and we determine the so-called (spectral) symbol describing the asymptotic spectrum. We also study the properties of the symbol, which turns out to be a $d$-variate function taking values in the space of $N({\boldsymbol p})\times N({\boldsymbol p})$ Hermitian matrices, where $N({\boldsymbol p})=\prod_{j=1}^d p_j$. Unlike the stiffness matrices coming from the ${\boldsymbol p}$-degree B-spline isogeometric analysis approximation of the same differential problem, where a unique $d$-variate real-valued function describes all the spectrum, here the spectrum is described by $N({\boldsymbol p})$ different functions, i.e., the $N({\boldsymbol p})$ eigenvalues of the symbol, which are well-separated, far away, and exponentially diverging with respect to ${\boldsymbol p}$ and $d$. This very involved picture provides an explanation of: (a) the difficulties encountered in designing robust solvers, with convergence speed independent of the matrix size, of the approximation parameters ${\boldsymbol p}$, and of the dimensionality $d$; (b) the convergence deterioration of known iterative methods, already observed in practice for moderate ${\boldsymbol p}$ and $d$. (The PDF has been changed.)

Compact local integrated-RBF approximations for second-order elliptic differential problems

This paper presents a new compact approximation method for the discretisation of second-order elliptic equations in one and two dimensions. The problem domain, which can be rectangular or non-rectangular, is represented by a Cartesian grid. On stencils, which are three nodal points for one-dimensional problems and nine nodal points for two-dimensional problems, the approximations for the field variable and its derivatives are constructed using integrated radial basis functions (IRBFs). Several pieces of information about the governing differential equation on the stencil are incorporated into the IRBF approximations by means of the constants of integration. Numerical examples indicate that the proposed technique yields a very high rate of convergence with grid refinement.

Simulation of a nonlinear Steklov eigenvalue problem using finite-element approximation

Elliptic problems with parameters in the boundary conditions are called Steklov problems. With the tool of computational approximation (finite-element method), we estimate the solution of a nonlinear Steklov eigenvalue problem for a second-order, self-adjoint, elliptic differential problem. We discussed the behavior of the nonlinear problem with the help of computational results using Matlab.

A Cartesian-Grid Discretisation Scheme Based on Local Integrated RBFNs for Two-Dimensional Elliptic Problems

This paper reports a new numerical scheme based on Cartesian grids and local integrated radial-basis-function networks (IRBFNs) for the solution of second-order elliptic differential problems defined on two-dimensional regular and irregular domains. At each grid point, only neighbouring nodes are activated to construct the IRBFN approximations. Local IRBFNs are introduced into two different schemes for discretisation of partial differential equations, namely point collocation and control-volume (CV)/subregion-collocation. Linear (e.g. heat flow) and nonlinear (e.g. lid-driven triangular-cavity fluid flow) problems are considered. Numerical results indicate that the local IRBFN CV scheme outperforms the local IRBFN point-collocation scheme regarding accuracy. Moreover, the former shows a similar level of the matrix condition number and a significant improvement in accuracy over a linear CV method.

A numerical study of 2D integrated RBFNs incorporating Cartesian grids for solving 2D elliptic differential problems

AbstractThis article reports a numerical discretization scheme, based on two‐dimensional integrated radial‐basis‐function networks (2D‐IRBFNs) and rectangular grids, for solving second‐order elliptic partial differential equations defined on 2D nonrectangular domains. Unlike finite‐difference and 1D‐IRBFN Cartesian‐grid techniques, the present discretization method is based on an approximation scheme that allows the field variable and its derivatives to be evaluated anywhere within the domain and on the boundaries, regardless of the shape of the problem domain. We discuss the following two particular strengths, which the proposed Cartesian‐grid‐based procedure possesses, namely (i) the implementation of Neumann boundary conditions on irregular boundaries and (ii) the use of high‐order integration schemes to evaluate flux integrals arising from a control‐volume discretization on irregular domains. A new preconditioning scheme is suggested to improve the 2D‐IRBFN matrix condition number. Good accuracy and high‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

Mixed formulations for a class of variational inequalities

A general setting is proposed for the mixed finite element approximations of elliptic differential problems involving a unilateral boundary condition. The treatment covers the Signorini problem as well as the unilateral contact problem with or without friction. Existence, uniqueness for both the continuous and the discrete problem as well as error estimates are established in a general framework. As an application, the approximation of the Signorini problem by the lowest order mixed finite element method of Raviart–Thomas is proved to converge with a quasi-optimal error bound.

Multi-iterative methods

A general class of iterative methods is introduced for solving positive definite linear systems Ax = b. These methods use two or more different iterative techniques and each of them reduces the error by a constant factor in a different subspace of R n . The Multigrid, as an example of a multi-iterative method, combines these partial results and we reach a reduction of the error by θ ν at each step, where ν is the number of applications of the first scheme and θ \\gt 1 is a constant independent of n. In the second part, we consider the “prolongation” operator p with a special “antidiagonal” structure and we show that the multigrid method for Toeplitz matrices [1] is an example of a method of this class. In a third part, we extend the multigrid method to block-Toeplitz matrices, obtaining a “block-antidiagonal” structure; for Toeplitz and block-Toeplitz cases, the choice of p which maximizes convergence speed simply depends on the coefficients of matrix A. The numerical results on 1-D and 2-D elliptic differential problems confirm the effectiveness of this multi-iterative method; in fact its spectral radius is very small and independent of the mesh spacing h.

A multigrid method for grid generation with line-spacing control

A multigrid method for grid generation on two-dimensional regions and its applications to test problems are presented. The multigrid algorithm deals with the solution of elliptic differential problems which occur in the computation of boundary-fitted grids. The solution of elliptic systems of partial differential equations, which correspond to transformed Poisson systems, is carried out by a full approximation storage (FAS) algorithm. The components of the method, such as the relaxation for error smoothing and the coarsening strategy, are evaluated on problems in which sources of attractions are considered, and the generated grids are shown by figures.

Elliptic Differential Problems with High Order Boundary Conditions.

Elliptic Differential Problems with High Order Boundary Conditions.