Narrow Results

In this paper, a novel isogeometric method for Biot’s consolidation model is constructed and analyzed, using a four-field formulation where the unknown variables are the solid displacement, solid pressure, fluid flux, and fluid pressure. Mixed isogeometric spaces based on B-splines basis functions are employed in the space discretization, allowing a smooth representation of the problem geometry and solution fields. The main result of the paper is the proof of optimal error estimates that are robust with respect to material parameters for all solution fields, particularly in the case of nearly incompressible materials. The analysis does not require a uniformly positive storage coefficient. The results of numerical experiments in two and three dimensions confirm the theoretical error estimates and high-order convergence rates attained by the proposed isogeometric Biot discretization and assess its performance with respect to the mesh size, spline polynomial degree, spline regularity, and material parameters.

The purpose of this paper is to present a rather simple proof of an inequality of Nečas⁹ which is equivalent to the inf–sup condition. This inequality is fundamental in the study of the Stokes equations. The boundary of the domain is only assumed to be Lipschitz.

This paper is devoted to the setting and analysis of a Petrov–Galerkin wavelet-fictitious domain numerical method for the approximation of the solution of multi-dimensional parabolic equations on a general domain.

In this method, the original parabolic equation, set on a domain ω, is first discretized in time using a finite difference scheme. At each time step, the corresponding elliptic equation on ω is transformed into a saddle point problem on a functional space defined on a bigger but simply shaped domain Ω where the initial boundary conditions on ω are enforced using surface Lagrange multipliers (Ref. 2). The solution of this problem is then approximated, thanks to a Petrov–Galerkin formulation, using wavelets and time scheme associated "vaguelettes" (Ref. 8).

Existence, uniqueness and convergence of the approximated solution, when finite dimension spaces are used, are established and the efficiency as well as the stability of the numerical algorithm (namely the Uzawa algorithm) used in the resolution are analyzed. The constraint of a discrete inf-sup condition as well as ill-conditioning associated to the trace operator are investigated in the wavelet framework.

Numerical results related to the 2D heat equation are presented.

In a three-dimensional bounded possibly multiply connected domain, we give gradient and higher-order estimates of vector fields via div and curl in L^p-theory. Then, we prove the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, we consider the stationary Stokes equations with non-standard boundary conditions of the form u × n = g × n and π = π₀ on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions. Our proofs are based on obtaining Inf–Sup conditions that play a fundamental role. We give a variant of the Stokes system with these boundary conditions, in the case where the compatibility condition is not verified. Finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as u⋅n and u × n on Γ.

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order $k$ from [Finite element approximations of symmetric tensors on simplicial grids in ${ℝ}^n$: The higher order case, J. Comput. Math.33 (2015) 283–296], enriched, for each $(n-1)$-dimensional simplex, by $\frac{(n+1)n}{2}$ face bubble functions in the symmetric tensor finite element space of order $n+1$ from [Finite element approximations of symmetric tensors on simplicial grids in ${ℝ}^n$: The higher order case, J. Comput. Math.33 (2015) 283–296] when $1\le k\le n-1$, and by $\frac{(n-1)n}{2}$ face bubble functions in the symmetric tensor finite element space of order $n+1$ from [Finite element approximations of symmetric tensors on simplicial grids in ${ℝ}^n$: The higher order case, J. Comput. Math.33 (2015) 283–296] when $k=n$. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise $P_{k-1}$ polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element $18$ plus $3$ in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput.77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math.92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

In this paper, a new mixed finite element scheme is given based on the less regularity of velocity for the single phase compressible flow in practice. Based on the new mixed variational formulation, we give its stable conforming finite element approximation for the P₀–P₁ pair and its stabilized conforming finite element approximation for the P₁–P₁ pair. Moreover, optimal error estimates are derived in H¹-norm and L²-norm for the approximation of pressure and error estimate in L²-norm for the approximation of velocity by using two methods. Finally, numerical tests confirm the theoretical results of our methods.