A finite element method for a unidimensional single-phase nonlinear free boundary problem in groundwater flow

Consider a unidimensional, single-phase nonlinear Stefan problemwith nonlinear source and permeance terms, and a Dirichlet boundarycondition depending on the free boundary function. This problemis important in groundwater flow. By immobilizing the free boundarywith the help of a Landau-type transformation, together witha homogeneous transformation dealing with the nonhomogeneousDirichlet boundary condition, an H¹-finite element method forthe problem is proposed and analyzed. Global existence of theapproximate solution is established, and optimal error estimatesin L², L, H¹ and H² norms are derived for both semi-discreteand fully discrete schemes.     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

Viscous flow through corrugated tube by boundary element method

Summary The creeping flow of a Newtonian fluid through a sinusoidally-corrugated tube is solved by the Boundary Element Method. Agreement with another numerical method is noted. In addition, it is shown that previous perturbation theory is valid only when the corrugation amplitude is small (<0.3a) and the wavelength of the corrugation is large (>3a), wherea is the mean radius of the tube.

---

Zusammenfassung Das Problem der schleichenden Bewegung eines Newton'schen Fluids durch ein Rohr mit sinusförmig gewellter Wand wird mit Hilfe der Boundary Element-Methode gelöst. Übereinstimmung mit einer anderen numerischen Methode wird festestellt. Zudem wird gezeigt, daß eine früher gefundene Störungstheorie nur gültig ist wenn die Wellenamplitude klein (<0.3a) und die Wellenlänge groß (>3a) ist (a=mittlerer Rohrradius).

     

A finite element method for a noncoercive elliptic problem with Neumann boundary conditions

We consider a noncoercive convection-diffusion problem with Neumann boundary conditions appearing in modeling of magnetic fluid seals. The associated operator has a non-trivial one-dimensional kernel spanned by a positive function. A discretization is proposed preserving these properties. Optimal error estimates in the H¹-norm are based on a discrete stability result. Numerical results confirm the theoretical predictions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discontinuous Galerkin finite element method for a nonlinear boundary value problem

In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We show the optimal error estimates in the DG-norm （stronger than the H1 norm） and the L2 norm, respectively. Furthermore, some residual-based a posteriori error estimators are established which provide global upper bounds and local lower bounds on the discretization error. These a posteriori analysis results can be applied to develop the adaptive DG methods.     

A mixed finite element method for the heat flow problem

A semidiscrete finite element scheme for the approximation of the spatial temperature change field is presented. The method yields a better order of convergence than the conventional use of linear elements.     

A finite element method with a large mesh-width for a stiff two-point boundary value problem

We describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed two-point boundary value problem on [?1, 1]

$\text{? u″ + pu′ = f, u(?1) = u(1) = 0, 0 < ? ∠ 1, (′ =}\text{d}\text{dx}\text{)}$

On a quasi-uniform mesh we construct exponentially fitted trial spaces which consist of piece-wise polynomials and of exponentials which fit locally to the singular solution of the equation or its adjoint. We discretise the Galerkin form for the boundary problem using such exponentially fitted trial spaces. We derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the mesh-points, the error depending on ?, the mesh-width and the degree of the piece-wise polynomials.     

A mixed finite element method for a quasi‐Newtonian fluid flow

We propose a mixed formulation for quasi‐Newtonian fluid flow obeying the power law where the stress tensor is introduced as a new variable. Based on such a formulation, a mixed finite element is constructed and analyzed. This finite element method possesses local (i.e., at element level) conservation properties (conservation of the momentum and the mass) as in the finite volume methods. We give existence and uniqueness results for the continuous problem and its approximation and we prove error bounds. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

The finite element method for a boundary value problem with strong singularity

The existence and uniqueness of the R_ν-generalized solution for the third-boundary-value problem and the non-self-adjoint second-order elliptic equation with strong singularity are established. We construct a finite element method with a basis containing singular functions. The rate of convergence of the approximate solution to the R_ν-generalized solution in the norm of the Sobolev weighted space is established and, finally, results of numerical experiments are presented.     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

A modified scaled boundary finite element method for dynamic response of a discontinuous layered half-space

In this paper, a modified scaled boundary finite element method is proposed to deal with the dynamic analysis of a discontinuous layered half-space. In order to describe the geometry of discontinuous layered half-space exactly, splicing lines, rather than a point, are chosen as the scaling center. Based on the modified scaled boundary transformation of the geometry, the Galerkin's weighted residual technique is applied to obtain the corresponding scaled boundary finite element equations in displacement. Then a modified version of dimensionless frequency is defined, and the governing first-order partial differential equations in dynamic stiffness with respect to the excitation frequency are obtained. The global stiffness is obtained by adding the dynamic stiffness of the interior domain calculated by a standard finite element method, and the dynamic stiffness of far field is calculated by the proposed method. The comparison of two existing solutions for a horizontal layered half-space confirms the accuracy and efficiency of the proposed approach. Finally, the dynamic response of a discontinuous layered half-space due to vertical uniform strip loadings is investigated.     

Convergence of a finite element method for non-parametric mean curvature flow

Summary. Convergence for the spatial discretization by linear finite elements of the non-parametric mean curvature flow is proved under natural regularity assumptions on the continuous solution. Asymptotic convergence is also obtained for the time derivative which is proportional to mean curvature. An existence result for the continuous problem in adequate spaces is included. Received September 30, 1993     

A direct discontinuous Galerkin finite element method for convection-dominated two-point boundary value problems

The direct discontinuous Galerkin (DDG) finite element method, using piecewise polynomials of degree k ≥ 1 on a Shishkin mesh, is applied to convection-dominated singularly perturbed two-point boundary value problems. Consistency, stability and convergence of order k (up to a logarithmic factor) are proved in an energy-type norm appropriate to the method and problem. The results are robust, i.e., they hold uniformly for all values of the singular perturbation parameter. Numerical experiments confirm the theoretical convergence rate.

     

A coupled finite and boundary spectral element method for linear water-wave propagation problems

A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22–34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre–Gauss–Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p-convergence of spectral methods.     

The Galerkin boundary element method for transient Stokes flow

Since the fundamental solution for transient Stokes flow in three dimensions is complicated it is difficult to implement discretization methods for boundary integral formulations. We derive a representation of the Stokeslet and stresslet in terms of incomplete gamma functions and investigate the nature of the singularity of the single- and double layer potentials. Further, we give analytical formulas for the time integration and develop Galerkin schemes with tensor product piecewise polynomial ansatz functions. Numerical results demonstrate optimal convergence rates.     

Preconditioners for the p-version of the Galerkin method for a coupled finite element/boundary element system

We propose and analyze efficient preconditioners for solving systems of equations arising from the p-version for the finite element/boundary element coupling. The first preconditioner amounts to a block Jacobi method, whereas the second one is partly given by diagonal scaling. We use the generalized minimum residual method for the solution of the linear system. For our first preconditioner, the number of iterations of the GMRES necessary to obtain a given accuracy grows like log² p, where p is the polynomial degree of the ansatz functions. The second preconditioner, which is more easily implemented, leads to a number of iterations that behave like p log³ p. Computational results are presented to support this theory. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 47–61, 1998     

Solving parabolic Wick-stochastic boundary value problems using a finite element method

A class of linear parabolic stochastic boundary value problems of Wick-type is studied. The equations are understood in a weak sense on a suitable stochastic distribution space, and existence and uniqueness results are provided. The paper continues to discuss a numerical method for this type of problem, based on a Galerkin type of approximation. Estimates showing linear convergence in time and space are derived, and rate of convergence results for the stochastic dimension are reported.     

A finite element method for a single phase quasilinear free boundary problem in one space dimension

In this paper, we apply finite element Galerkin method to a singlephase quasilinear Stefan problem with a forcing term. To construct the fully discrete approximation we apply the extrapolated Crank-Nicolson method and we derive the optimal order of convergence 2 in the temporal direction inL ²,H ¹ normed spaces.