Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

/pdf/asymptotic-analysis-of-periodic-structures-14wjonu4ba.pdf

Asymptotic Analysis of Periodic Structures

Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.

/pdf/homogenization-and-two-scale-convergence-475vs128x8.pdf

Homogenization and two-scale convergence

http://www.cmap.polytechnique.fr/~jouve/papers/jcp_final.pdf

Structural optimization using sensitivity analysis and a level-set method

Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.

Fine phase mixtures as minimizers of energy

Definition and weak stability of nonconservative products

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/pdf/compacite-par-compensation-197icvqxz7.pdf

Compacité par compensation

/pdf/compacite-par-compensation-2eq3dcp0ki.pdf

We study the nonlinear monotone elliptic problem  −div (a(x,∇u)) = μ on Ω, u = 0 on ∂Ω, when Ω ⊂ R , μ is a Radon measure with bounded total variation on Ω, 1 < p ≤ N , and u 7→ −div (a(x,∇u)) is a monotone operator acting on W 1,p 0 (Ω). We introduce a new definition of solution (the renormalized solution) in four equivalent ways. We prove the existence of a renormalized solution by an approximation procedure, where the key point is a stability result (the strong convergence in W 1,p 0 (Ω) of the truncates). We also prove partial uniqueness results. SISSA, via Beirut 4, 34014 Trieste, Italy Laboratoire d’Analyse Numerique, Universite Paris VI, Case 187, 75252 Paris cedex 05, France Dipartimento di Matematica, Universita di Roma I, P.le A. Moro 2, 00185 Roma, Italy Universite d’Orleans, rue de Chartres, B.P. 6759, 45067 Orleans cedex 2, France

/pdf/renormalized-solutions-of-elliptic-equations-with-general-2pld7ewfok.pdf

Renormalized solutions of elliptic equations with general measure data

https://people.maths.ox.ac.uk/~ball/Papers/Ball%20&%20Murat%201984.pdf

François Murat

Papers

Definition and weak stability of nonconservative products

Compacité par compensation

Compacité par compensation

Renormalized solutions of elliptic equations with general measure data

W1,p-quasiconvexity and variational problems for multiple integrals