A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

In this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.

This paper develops a framework to include Dirichlet boundary conditions on a subset ofthe boundary which depends on time. In this model, the boundary conditions are weaklyenforced with the help of a Lagrange multiplier method. In order to avoid that the ansatzspace of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, whichmaps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition aswell as existence results are presented for a class of second order initial-boundary valueproblems. For the semi-discretization in space, a finite element scheme is presented whichsatisfies a discrete stability condition. Because of the saddle point structure of theunderlying PDE, the resulting system is a DAE of index 3.

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair P1 — P1 which do not satisfy the inf-sup condition. The two-scale method consist of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the H1-norm for velocity and the L2-norm for pressure are obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation h = 𝾪 (H2). Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.

A new approach for computationally efficient estimation of stability factors forparametric partial differential equations is presented. The general parametric bilinearform of the problem is approximated by two affinely parametrized bilinear forms atdifferent levels of accuracy (after an empirical interpolation procedure). The successiveconstraint method is applied on the coarse level to obtain a lower bound for the stabilityfactors, and this bound is extended to the fine level by adding a proper correction term.Because the approximate problems are affine, an efficient offline/online computationalscheme can be developed for the certified solution (error bounds and stability factors) ofthe parametric equations considered. We experiment with different correction terms suitedfor a posteriori error estimation of the reduced basis solution ofelliptic coercive and noncoercive problems.

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods fornearly and perfectly incompressible linear elasticity. These mixed methods allow thechoice of polynomials of any order k ≥ 1 for the approximation of thedisplacement field, and of order k or k − 1 for thepressure space, and are stable for any positive value of the stabilization parameter. Weprove the optimal convergence of the displacement and stress fields in both cases, witherror estimates that are independent of the value of the Poisson’s ratio. These estimatesdemonstrate that these methods are locking-free. To this end, we prove the correspondinginf-sup condition, which for the equal-order case, requires a construction to establishthe surjectivity of the space of discrete divergences on the pressure space. In theparticular case of near incompressibility and equal-order approximation of thedisplacement and pressure fields, the mixed method is equivalent to a displacement methodproposed earlier by Lew et al. [Appel. Math. Res. express3 (2004) 73–106]. The absence of locking of this displacementmethod then follows directly from that of the mixed method, including the uniform errorestimate for the stress with respect to the Poisson’s ratio. We showcase the performanceof these methods through numerical examples, which show that locking may appear ifDirichlet boundary conditions are imposed strongly rather than weakly, as we do here.

A new solution methodology is proposed for solving efficiently Helmholtz problems. The proposed method falls in the category of the discontinuous Galerkin methods. However, unlike the existing solution methodologies, this method requires solving (a) well-posed local problems to determine the primal variable, and (b) a global positive semi-definite Hermitian system to evaluate the Lagrange multiplier needed to restore the continuity across the element edges. Illustrative numerical results obtained for two-dimensional interior Helmholtz problems are presented to assess the accuracy and the stability of the proposed solution methodology.

Liquid crystal elastomers present features not found in ordinary elastic materials, such as semi-soft elasticity and the related stripe domain phenomenon. In this paper, the two-dimensional Bladon–Terentjev–Warner model and the one-constant Oseen–Frank energy expression are combined to study the liquid crystal elastomer. We also impose two material constraints, the incompressibility of the elastomer and the unit director norm of the liquid crystal. We prove existence of minimiser of the energy for the proposed model. Next we formulate the discrete model, and also prove that it possesses a minimiser of the energy. The inf-sup values of the discrete linearised system are then related to the smallest singular values of certain matrices. Next the existence and uniqueness of the Lagrange multipliers associated with the two material constraints are proved under the assumption that the inf-sup conditions hold. Finally numerical simulations of the clamped-pulling experiment are presented for elastomer samples with aspect ratio 1 or 3. The semi-soft elasticity is successfully recovered in both cases. The stripe domain phenomenon, however, is not observed, which might be due to the relative coarse mesh employed in the numerical experiment. Possible improvements are discussed that might lead to the recovery of the stripe domain phenomenon.

We consider a family of quadrilateral or hexahedral mixed hp-finite elements for an incompressible flow problem with Q r -elements for the velocity anddiscontinuous $P_{r-1}$ -elements for the pressure where the orderr can vary from element to elementbetween 2 and an arbitrary bound.For multilevel adaptive grids with hanging nodes and a sufficiently small mesh size,we prove the inf-sup condition uniformly with respect to the meshsize and the polynomial degree.

We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Q r -elements for the velocity anddiscontinuous $P_{r-1}$ -elements for the pressure where the orderr can vary from element to elementbetween 2 and a fixed bound $r^*$ .We prove the inf-sup condition uniformly with respect to the meshwidth hon general quadrilateral and hexahedral meshes with hanging nodes.

The present paper proposes and analyzes a general locking free mixed strategy for computing the deformation of incompressible three dimensional structures placed inside flexible membranes. The model involves as in Chapelle and Ferent [Math. Models Methods Appl. Sci.13 (2003) 573–595] a bending dominated shell envelope and a quasi incompressible elastic body. The present work extends an earlier work ofArnold and Brezzi [Math Comp.66 (1997) 1–14] treating the shell part and proposes a global stable finite element approximation by coupling optimal mixed finite element formulations of the different subproblems by mortar techniques. Examples of adequate finite elements are proposed. Convergence results are derived in two steps. First a global inf-sup condition is proved, deducedfrom the local conditions to be satisfied by the finite elements used for the external shell problem, the internal incompressible 3D problem, and the mortar coupling, respectively. Second, the analysis ofArnold and Brezzi [Math. Comp.66 (1997) 1–14] is extended to the present problem and least to convergence results for the full coupled problem, withconstants independent of the problem's small parameters.

We propose and analyze several finite-element schemes for solving a grade-twofluid model, with atangential boundary condition, in a two-dimensional polygon. The exactproblem is split into ageneralized Stokes problem and a transport equation, in such a way that italways has a solutionwithout restriction on the shape of the domain and on the size of the data.The first scheme usesdivergence-free discrete velocities and a centered discretization of thetransport term, whereas theother schemes use Hood-Taylor discretizations for the velocity andpressure, and either a centered or an upwinddiscretization of the transport term. One facet of our analysis isthat, without restrictions on the data,each scheme has a discrete solution and all discrete solutions convergestrongly to solutions of theexact problem. Furthermore, if the domain is convex and the data satisfycertain conditions, eachscheme satisfies error inequalities that lead to error estimates.

For the Stokes problem in a two- or three-dimensionalbounded domain, we propose a new mixed finite element discretization which relies ona nonconforming approximation of the velocity and a more accurate approximation of thepressure. We prove that the velocity and pressure discrete spaces are compatible, in thesense that they satisfy an inf-sup condition of Babuška and Brezzi type, and wederive some error estimates.