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A numerical method for simulating transient ows of gas-liquid mixtures is proposed. The mathematical model, established for a suspension of gas bubbles in liquid, includes an equation taking into account the relative velocity between the gas and liquid. A numerical technique based on the MacCormack scheme combined with the method of characteristics for the boundary conditions is presented. Theoretical results for transients initiated by a rapid closing valves are compared with measure- ments. A good agreement is found particularly for large values of initial dissolved gas concentration. [ABSTRACT FROM AUTHOR]

In this article we study a projection-stabilized nonconforming finite element discretization of the Stokes problem. We present a priori error analysis and give a recovery-based a posteriori error estimator for the considered problem. Numerical results illustrate the theoretical performance of the error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 218-240, 2017 [ABSTRACT FROM AUTHOR]

We develop in this article an a posteriori error estimator for the P1-nonconforming finite element approximation, for a diffusion-reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950-976, 2015 [ABSTRACT FROM AUTHOR]

We consider a recently proposed finite-element space that consists of piecewise affine functions with discontinuities across a smooth given interface Γ (a curve in two dimensions, a surface in three dimensions). Contrary to existing extended finite element methodologies, the space is a variant of the standard conforming ℙ1 space that can be implemented element by element. Further, it neither introduces new unknowns nor deteriorates the sparsity structure. It is proved that, for u arbitrary in W1,p(Ω∖Γ)∩W2,s(Ω∖Γ), the interpolant ℐhu defined by this new space satisfieswhere h is the mesh size, Ω⊂ℝd is the domain, p>d, p≥q, s≥q and standard notation has been adopted for the function spaces. This result proves the good approximation properties of the finite-element space as compared to any space consisting of functions that are continuous across Γ, which would yield an error in the Lq(Ω)-norm of order . These properties make this space especially attractive for approximating the pressure in problems with surface tension or other immersed interfaces that lead to discontinuities in the pressure field. Furthermore, the result still holds for interfaces that end within the domain, as happens for example in cracked domains. [ABSTRACT FROM PUBLISHER]

In this Note, we show that modification of Bank–Wieser estimator introduce an L∞-a posteriori error estimator for conforming and nonconforming methods. We prove, without saturation assumption nor comparison with residual estimators, the equivalence with the L∞ error. To cite this article: A. Agouzal, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 411–415. [Copyright &y& Elsevier]

In this Note, we show that a modified and simplified version of the estimator of Bank–Weiser can be used to define a robust a posteriori error estimator for singularly perturbed problem. We prove without comparison with a residual estimator or saturation assumption, the equivalence of the estimator with the error in the energy norm and the robusteness with respect to the diffusion coefficient. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]