Parabolic finite volume element equations in nonconvex polygonal domains

We study spatially semidiscrete and fully discrete finite volume element approximations of the heat equation with homogeneous Dirichlet boundary conditions in a plane polygonal domain with one reentrant corner. We show that, as a result of the singularity in the solution near the reentrant corner, the convergence rate is reduced from optimal second order, similarly to what was shown for the finite element method in the earlier work 2 . Optimal order convergence may be restored by mesh refinement near the corners of the domain. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

FINITE ELEMENT ANALYSIS OF FREE VIBRATION OF LIQUID－FILLED AXISYMMETRIC TANKS

ＦＵＲＵＤＥ（符儒德）；ＵＳＨＩＪＩＭＡＴＥＲＵＯ（牛岛照夫）（ＤｅｐａｒｔｍｅｎｔｏｆＣｏｍｐｕｔｅｒＳｃｉｅｎｃｅ＆ＩｎｆｏｒｍａｔｉｏｎＭａｔｈｅｍａｔｉｃｓ，ＵｎｉｖｅｒｓｉｔｙｏｆＥｌｅｃｔｒｏ－Ｃｏｍｍｕｎｉｃａｔｉｏｎｓ，Ｔｏｋｙｏ１８...     

Mortar spectral element discretization of the stokes problem in axisymmetric domains

The Stokes problem in a tri‐dimensional axisymmetric domain results into a countable family of two‐dimensional problems when using the Fourier coefficients with respect to the angular variable. Relying on this dimension reduction, we propose and study a mortar spectral element discretization of the problem. Numerical experiments confirm the efficiency of this method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 44–73, 2014     

Finite element solution of nonlinear diffusion problems

In this paper we describe the Rothe-finite element numerical scheme to find an approximate solution of a nonlinear diffusion problem modeled as a parabolic partial differential equation of even order. This scheme is based on the Rothe’s approximation in time and on the finite element method (FEM) approximation in the spatial discretization. A proof of convergence of the approximate solution is given and error estimates are shown.     

Finite element solutions for three‐dimensional elliptic boundary value problems on unbounded domains

The finite element (FE) solutions of a general elliptic equation ?div([a_ij] ??u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R ³, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω_∞, so that the remaining part Ω_B = Ω\Ω _∞ is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γ_a = Ω _∞ ∩ Ω _B. In this article, instead of discarding an unbounded subdomain Ω_∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babu?ka and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Improving the Rate of Convergence of High-Order Finite Elements on Polyhedra I: A Priori Estimates

ABSTRACT

Let 𝒯 _k be a sequence of triangulations of a polyhedron Ω ? ? ⁿ and let S _k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u _k  ∈ S _k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S _k ensures optimal rates of convergence for the sequence u _k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet–Neumann boundary conditions on a polygon.     

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L₂ and H 1 . The convergence rate in the L_∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation

In this paper, a new one-dimensional space-fractional Boussinesq equation is proposed. Two novel numerical methods with a nonlocal operator (using nodal basis functions) for the space-fractional Boussinesq equation are derived. These methods are based on the finite volume and finite element methods, respectively. Finally, some numerical results using fractional Boussinesq equation with the maximally positive skewness and the maximally negative skewness are given to demonstrate the strong potential of these approaches. The novel simulation techniques provide excellent tools for practical problems. These new numerical models can be extended to two- and three-dimensional fractional space-fractional Boussinesq equations in future research where we plan to apply these new numerical models for simulating the tidal water table fluctuations in a coastal aquifer.     

On error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains

Numerical verification methods, so-called Nakao's methods, on existence or uniqueness of solutions to PDEs have been developed by Nakao and his group including the authors. They are based on the error estimation of approximate solutions which are mainly computed by FEM.     

A complex variable boundary element method for axisymmetric heat conduction in a nonhomogeneous solid

A complex variable boundary element method is proposed for solving numerically the axisymmetric steady-state problem of heat conduction in a nonhomogeneous isotropic solid. To assess the validity and the accuracy of the method, it is applied to solve specific cases of the problem. The numerical solutions obtained agree well with known solutions.     

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.     

Finite element modeling of the nonlinear oscillations in axisymmetric acoustic resonators

Oscillation of a gas in closed resonators has gained considerable interest in the past years. In this paper, the nonlinear equations governing the behavior of the gas oscillations inside the resonator are formulated in a weak form and then modeled using the finite element method. The pressure ratios, predicted by the proposed model, are in close agreement with the exact solutions available for simple geometries such as cylindrical, exponential and linearly varying area resonators. The presented comparisons validate the accuracy of the finite element model and emphasize its potential for predicting the performance or resonators of more complex geometries which are necessary for generating high pressures from the standing waves. Also, gas flow through the boundaries of the resonator is implemented in the proposed model. The presented finite element model presents an invaluable tool for designing a new class of acoustic compressors which can be used, for example, in refrigeration and vibration control applications.     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.     

Finite element approximation of the elasticity spectral problem on curved domains

We analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non-convex domain. In the framework of the abstract spectral approximation theory, we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors. Two kinds of problems are considered: the discrete domain does not coincide with the real one and mixed boundary conditions are imposed. Some numerical results are presented.     

Finite element approximation of spectral problems with Neumann boundary conditions on curved domains

This paper deals with the finite element approximation of the spectral problem for the Laplace equation with Neumann boundary conditions on a curved nonconvex domain . Convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain in the framework of the abstract spectral approximation theory.

     

Solution of Poisson’s equation by analytical boundary element integration

The solution of Poisson’s equation is essential for many branches of science and engineering such as fluid-mechanics, geosciences, and electrostatics. Solution of two-dimensional Poisson’s equations is carried out by BEM based on Galerkin Vector Method in which the integrals that appear in the boundary element method are expressed by analytical integration. In this paper, the Galerkin vector method is developed for more general case than presented in the previous works. The integrals are computed for constant and linear elements in BEM. By employing analytical integration in BEM computation, the numerical schemes and coordinate transformations can be avoided. The presented method can also be used for the multiple domain case. The results of the analytical integration are employed in BEM code and the obtained analytical expression will be applied to several examples where the exact solution exists. The produced results are in good agreement with the exact solution.     

Convergence analysis of an upwind finite volume element method with Crouzeix-Raviart element for non-selfadjoint and indefinite problems

In this paper we construct an upwind finite volume element scheme based on the Crouzeix-Raviart nonconforming element for non-selfadjoint elliptic problems. These problems often appear in dealing with flow in porous media. We establish the optimal order H ¹-norm error estimate. We also give the uniform convergence under minimal elliptic regularity assumption      

The Neumann problem for the Laplace equation on general domains

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set G in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on ∂G. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on G a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.