Natural hp‐BEM for the electric field integral equation with singular solutions

We apply the hp ‐version of the boundary element method (BEM) for the numerical solution of the electric field integral equation (EFIE) on a Lipschitz polyhedral surface Γ. The underlying meshes are supposed to be quasi‐uniform triangulations of Γ, and the approximations are based on either Raviart‐Thomas or Brezzi‐Douglas‐Marini families of surface elements. Nonsmoothness of Γ leads to singularities in the solution of the EFIE, severely affecting convergence rates of the BEM. However, the singular behavior of the solution can be explicitly specified using a finite set of functions (vertex‐, edge‐, and vertex‐edge singularities), which are the products of power functions and poly‐logarithmic terms. In this article, we use this fact to perform an a priori error analysis of the hp ‐BEM on quasi‐uniform meshes. We prove precise error estimates in terms of the polynomial degree p, the mesh size h, and the singularity exponents. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015     

Preconditioning and a posteriori error estimates using h‐ and p‐hierarchical finite elements with rectangular supports

We show some of the properties of the algebraic multilevel iterative methods when the hierarchical bases of finite elements (FEs) with rectangular supports are used for solving the elliptic boundary value problems. In particular, we study two types of hierarchies; the so‐called h‐ and p‐hierarchical FE spaces on a two‐dimensional domain. We compute uniform estimates of the strengthened Cauchy–Bunyakowski–Schwarz inequality constants for these spaces. Moreover, for diagonal blocks of the stiffness matrices corresponding to the fine spaces, the optimal preconditioning matrices can be found, which have tri‐ or five‐diagonal forms for h‐ or p‐refinements, respectively, after a certain reordering of the elements. As another use of the hierarchical bases, the a posteriori error estimates can be computed. We compare them in test examples for h‐ and p‐hierarchical FEs with rectangular supports. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A layer adaptive B‐spline collocation method for singularly perturbed one‐dimensional parabolic problem with a boundary turning point

In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a₀(x, t)x^p, where a₀(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011     

Uniform approximation of singularly perturbed reaction‐diffusion problems by the finite element method on a Shishkin mesh

We consider the numerical approximation of singularly perturbed reaction‐diffusion problems over two‐dimensional domains with smooth boundary. Using the h version of the finite element method over appropriately designed piecewise uniform (Shishkin) meshes, we are able to uniformly approximate the solution at a quasi‐optimal rate. The results of numerical computations showing agreement with the analysis are also presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 89–111, 2003     

Infinite families of non‐embeddable quasi‐residual Menon designs

A Menon design of order h² is a symmetric (4h²,2h²‐h,h²‐h)‐design. Quasi‐residual and quasi‐derived designs of a Menon design have parameters 2‐(2h² + h,h²,h²‐h) and 2‐(2h²‐h,h²‐h,h²‐h‐1), respectively. In this article, regular Hadamard matrices are used to construct non‐embeddable quasi‐residual and quasi‐derived Menon designs. As applications, we construct the first two new infinite families of non‐embeddable quasi‐residual and quasi‐derived Menon designs. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 53–62, 2009     

Convergence analysis of an hp finite element method for singularly perturbed transmission problems in smooth domains

We consider a two‐dimensional singularly perturbed transmission problem with two different diffusion coefficients, in a domain with smooth (analytic) boundary. The solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high and interface layers along the interface. Utilizing existing and newly derived regularity results for the exact solution, we prove the robustness of an hp finite element method for its approximation. Under the assumption of analytic input data, we show that the method converges at an “exponential” rate, provided the mesh and polynomial degree distribution are chosen appropriately. Numerical results illustrating our theoretical findings are also included. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

An hp‐adaptive refinement strategy for hypersingular operators on surfaces

An adaptive refinement strategy for the hp‐version of the boundary element method with hypersingular operators on surfaces is presented. The error indicators are based on local projections provided by two‐level decompositions of ansatz spaces with additional bubble functions. Assuming a saturation property and locally quasi‐uniform meshes, efficiency and reliability of the resulting error estimator is proved. A second error estimator based on mesh refinement and overlapping decompositions that better fulfills the saturation property is presented. The performance of the algorithm and the estimators is demonstrated for a model problem. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 396–419, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10011     

Nonconforming finite element methods with subgrid viscosity applied to advection‐diffusion‐reaction equations

A nonconforming (Crouzeix–Raviart) finite element method with subgrid viscosity is analyzed to approximate advection‐diffusion‐reaction equations. The error estimates are quasi‐optimal in the sense that keeping the Péclet number fixed, the estimates are suboptimal of order in the mesh size for the L²‐norm and optimal for the advective derivative on quasi‐uniform meshes. The method is also reformulated as a finite volume box scheme providing a reconstruction formula for the diffusive flux with local conservation properties. Numerical results are presented to illustrate the error analysis. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Action functional and quasi‐potential for the burgers equation in a bounded interval

Consider the viscous Burgers equation u_t + f(u)_x = εu_xx on the interval [0,1] with the inhomogeneous Dirichlet boundary conditions u(t,0) = ρ₀, u(t,1) = ρ₁. The flux f is the function f(u) = u(1 − u), ε > 0 is the viscosity, and the boundary data satisfy 0 < ρ₀ < ρ₁ < 1. We examine the quasi‐potential corresponding to an action functional arising from nonequilibrium statistical mechanical models associated with the above equation. We provide a static variational formula for the quasi‐potential and characterize the optimal paths for the dynamical problem. In contrast with previous cases, for small enough viscosity, the variational problem defining the quasi‐potential admits more than one minimizer. This phenomenon is interpreted as a nonequilibrium phase transition and corresponds to points where the superdifferential of the quasi‐potential is not a singleton. © 2011 Wiley Periodicals, Inc.     

Numerical study of the HP version of mixed discontinuous finite element methods for reaction‐diffusion problems: The 1D case

This article studies the stability and convergence of the hp version of the three families of mixed discontinuous finite element (MDFE) methods for the numerical solution of reaction‐diffusion problems. The focus of this article is on these problems for one space dimension. Error estimates are obtained explicitly in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Extensive numerical results to show convergence rates in h and p of the MDFE methods are presented. Theoretical and numerical comparisons between the three families of MDFE methods are described. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 525–553, 2003     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Discretization of the Poisson equation with non‐smooth data and emphasis on non‐convex domains

Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non‐smooth such as if they are in only. For the method of transposition (sometimes called very weak formulation) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied. The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi‐uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to .© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1433–1454, 2016     

Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models

The parameters in the governing system of partial differential equations of multiple‐network poroelasticity models typically vary over several orders of magnitude, making its stable discretization and efficient solution a challenging task. In this paper, we prove the uniform Ladyzhenskaya–Babu?ka–Brezzi (LBB) condition and design uniformly stable discretizations and parameter‐robust preconditioners for flux‐based formulations of multiporosity/multipermeability systems. Novel parameter‐matrix‐dependent norms that provide the key for establishing uniform LBB stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ but also to all the other model parameters, such as the permeability coefficients K_i; storage coefficients $c_{p_i}$; network transfer coefficients β_{i j},i,j = 1,…,n; the scale of the networks n; and the time step size τ. Moreover, strongly mass‐conservative discretizations that meet the required conditions for parameter‐robust LBB stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm‐equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.     

Viscous compressible barotropic symmetric flows with free boundary under general mass force Part I: Uniform‐in‐time bounds and stabilization

We consider symmetric flows of a viscous compressible barotropic fluid with a free boundary, under a general mass force depending both on the Eulerian and Lagrangian co‐ordinates, with arbitrarily large initial data. For a general non‐monotone state function p, we prove uniform‐in‐time energy bound and the uniform bounds for the density ρ, together with the stabilization as t → ∞ of the kinetic and potential energies. We also obtain H¹‐stabilization of the velocity v to zero provided that the second viscosity is zero. For either increasing or non‐decreasing p, we study the L^λ‐stabilization of ρ and the stabilization of the free boundary together with the corresponding ω‐limit set in the general case of non‐unique stationary solution possibly with zones of vacuum. In the case of increasing p and stationary densities ρ_S separated from zero, we establish the uniform‐in‐time H¹‐bounds and the uniform stabilization for ρ and v. All these results are stated and mainly proved in the Eulerian co‐ordinates. They are supplemented with the corresponding stabilization results in the Lagrangian co‐ordinates in the case of ρ_S separated from zero. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of Two Linearized Problems Modeling Viscous Two‐Layer Flows

Two problems that appear in the linearization of certain free boundary value problems of the hydrodynamics of two viscous fluids are studied in the strip‐like domain Π = {x = (x₁, x₂) ∈ ℝ² : x₁ ∈ ℝ¹, (0 < x₂ < h_*) ∨ (h_* < x₂ < 1)}. The first problem arises in the linearization of a two‐layer flow down a geometrically perturbed inclined plane. The second one appears after the linearization of a two‐layer flow in a geometrically perturbed inclined channel with one moving (smooth) wall. For this purpose the unknown flow domain was mapped onto the double strip Π. The arising linear elliptic problems contain additional unknown functions in the boundary conditions. The paper is devoted to the investigation of these boundary problems by studying the asymptotics of the eigenvalues of corresponding operator pencils. It can be proved that the boundary value problems are uniquely solvable in weighted Sobolev spaces with exponential weight. The study of the full (nonlinear) free boundary value problems will be the topic of a forthcoming paper.     

A posteriori error estimation of the p‐Laplace problem

Two a posteriori error estimates are discussed for the p‐Laplace problem. Up to errors in their numerical computation, they provide a guaranteed upper bound for the W^1,p‐seminorm and a weighted W^1,2‐seminorm of u – u_h. The first, sharper a posteriori estimate is based on the numerical solution of local interface problems. The standard residual‐based error estimate is addressed with emphasis on involved constants, determined as local eigenvalues. Numerical examples that illustrate the performance of these estimators can be found in [3].     

Numerical solution to a one‐dimensional thermoplastic problem with unilateral constraint

In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ² + h²), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

The Crank‐Nicolson/Adams‐Bashforth scheme for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u₀ with div u₀ = 0, i.e., the time step τ satisfies: τ ≤ C₀ if u₀ ∈ H¹ ∩ L^∞; τ |log h| ≤ C₀ if u₀ ∈ H¹ for the mesh size h and some positive constant C₀. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012