Hybrid coupling of finite element and boundary element methods using Nitsche’s method and the Calderon projection

Numerical Algorithms - In this paper, we discuss a hybridised method for FEM-BEM coupling. The coupling from both sides use a Nitsche-type approach to couple to the trace variable. This leads to a...     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

Inverse problems and solution methods for a class of nonlinear complementarity problems

In this paper, motivated by the KKT optimality conditions for a sort of quadratic programs, we first introduce a class of nonlinear complementarity problems (NCPs). Then we present and discuss a kind of inverse problems of the NCPs, i.e., for a given feasible decision [`(x)]\bar{x} , we aim to characterize the set of parameter values for which there exists a point [`(y)]\bar{y} such that ([`(x)],[`(y)])(\bar{x},\bar{y}) forms a solution of the NCP and require the parameter values to be adjusted as little as possible. This leads to an inverse optimization problem. In particular, under ℓ _∞, ℓ ₁ and Frobenius norms as well as affine maps, this paper presents three simple and efficient solution methods for the inverse NCPs. Finally, some preliminary numerical results show that the proposed methods are very promising.     

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier–Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter \({\varepsilon}\), which means that the solution of the regularized problem converges to the solution of the Navier–Stokes type variational inequality problem as the parameter \({\varepsilon\longrightarrow 0}\). Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.     

Convergence analysis of finite element methods for H(curl; Ω)-elliptic interface problems

In this article we investigate the analysis of a finite element method for solving H(curl; ??)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nédélec H(curl; ??)-conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter ?? that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H(curl; ??)-norm are obtained for the first time. The analysis is based on a ??-strip argument, a new extension theorem for H ¹(curl; ??)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H(curl; ??)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.     

Adaptive methods for solvings minimax problems ∗

We consider finite-dimensional minimax problems for two traditional models: firstly,with box constraints at variables and,secondly,taking into account a finite number of tinear inequalities. We present finite exact primal and dual methods. These methods are adapted to a great extent to the specific structure of the cost function which is formed by a finite number of linear functions. During the iterations of the primal method we make use of the information from the dual problem, thereby increasing effectiveness. To improve the dual method we use the “long dual step” rule (the principle of ullrelaxation).The results are illustrated by numerical experiments.     

A posteriori error estimates of finite element method for parabolic problems

1IntroductionThebase0fadaPtivecomputing0ffiniteelementmethodisap0steri0rierr0restimates.I.Babuskaisthepioneerinthisfields.Manytechniquesaredevel0pedtoobtainaposteri0rierrorestimators.See[1-3,7-8,19-201.Theyaremainlybased0nthejumps0fthederiva-tivesontheboundariesoftl1eelel11elltandtheresidualintheelemellts.Recelltresultssh0wthatthereareveryclosedrelatiollsbetweellasymptoticexactap0steri0rierrorestimatesandsuperc0nvergence-SeealsoQ.Linetal.[11-13],andChen-Huang['].Therehasbeenmuchprogressill…     

Solution of double nonlinear problems in porous media by a combined finite volume–finite element algorithm

The combined finite volume–finite element scheme for a double nonlinear parabolic convection-dominated diffusion equation which models the variably saturated flow and contaminant transport problems in porous media is extended. Whereas the convection is approximated by a finite volume method (Multi-Point Flux Approximation), the diffusion is approximated by a finite element method. The scheme is fully implicit and involves a relaxation-regularized algorithm. Due to monotonicity and conservation properties of the approximated scheme and in view of the compactness theorem we show the convergence of the numerical scheme to the weak solution. Our scheme is applied for computing two dimensional examples with different degrees of complexity. The numerical results demonstrate that the proposed scheme gives good performance in convergence and accuracy.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

Boundary integral operators and boundary value problems for Laplace’s equation

In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with d-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.     

A modified weak Galerkin finite element methods for convection–diffusion problems in 2D

In this paper, we develop a modified weak Galerkin finite element method on arbitrary grids for convection–diffusion problems in two dimensions based on our previous work (Wang et al., J Comput Appl Math 271, 319–327, 2014), in which we only considered second order Poisson equations and thus only introduced a modified weak gradient operator. This method, called MWG-FEM, is based on a modified weak gradient operator and weak divergence operator which is put forward in this paper. Optimal order error estimates are established for the corresponding MWG-FEM approximations in both a discrete \(H^1\) norm and the standard \(L^2\) norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the MWG-FEM.     

The Dirichlet boundary value problems for p-Schrödinger operators on finite networks

In this paper, we define the discrete p-Schrödinger operators on finite networks and discuss the existence of the Dirichlet eigenvalues and their eigenfunctions for the operators. We also provide various equivalent conditions for the existence of positive solutions for Dirichlet boundary value problems for the operators.     

L∞-boundedness of the finite element galerkin operator for parabolic problems

In this paper the heat equation with Dirichlet boundary conditions in N ≤ 3 space dimensions - serving as model problem of second order parabolic initial boundary value problems - is considered. We prove: The standard finite element method is uniformly bounded in L_∞ with respect to space and time if the underlying finite elements are at least cubics.     

Sample solutions of stochastic boundary value problems ∗

We prove existence theorems for nonlinear stochastic Sturm-Liouville problems which improve results from [4]. In the simplest case this is done by means of a known result about measurable selections of multivalued maps and a new fixed point theorem for stochastic nonlinear operators which is more realistic than existing ones     

High order finite volume methods for singular perturbation problems

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.     

Optimal L∞-error estimates for nonconforming and mixed finite element methods of lowest order

Summary For second order linear elliptic problems, it is proved that theP ₁-nonconforming finite element method has the sameL -asymptotic accuracy as theP ₁-conforming one. This result is applied to derive optimalL -error estimates for both the displacement and the stress fields of the lowest order Raviart-Thomas mixed finite element method, and a superconvergence result at the barycenter of each element.Performed in the research program of Istituto di Analisi Numerica of C.N.R. of PaviaPartially supported by MPI, GNIM of CNR, ItalySupported by Consejo Nacional de Investigaciones Cientificas y Técnicas, Argentina