[PAST EVENT] Mathematics Colloquium: Thinh T. Kieu, Texas Tech University

Friday, November 22nd 2013
3pm - 4pm
Jones Hall, Room 302
200 Ukrop Way
Williamsburg, VA 23185
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Generalized Forchheimer Equations in Porous Media and Finite Element Approximation of Wave Equations
Full Description
This talk discusses several problems. The first part is devoted to Forchheimer flows in porous media for both single and multi phase problems. The second part is the finite element approximation of wave equations. Forchheimer flows. We generalize Darcy's law and common Forchheimer's laws to model nonlinear flows in porous media. For slightly compressible fluids, the problem is reduced to a
degenerate parabolic equation for the pressure. The solutions are estimated, particularly for large time, in supremum norm and different Sobolev norms. They are showed to be continuously dependent on the initial and boundary data, and the Forchheimer polynomials. Various techniques from partial differential equations and dynamical systems are used and combined with the special structure of the equation in our analysis.

We model multi-dimensional two-phase flows of incompressible fluids in porous media using generalized Forchheimer equations and the capillary pressure. We find a family of steady state solutions with certain geometric properties. To study their stability, the linearized system is derived and reduced to a parabolic equation for the saturation. This equation has a special structure depending on the steady states which we exploit to prove the stability results as well as the solutions' qualitative properties.

Numerical analysis. We investigate the mixed finite element spaces in discretization of acoustic wave equations. We apply mixed finite element approximations to the first-order form of the acoustic wave equation. Our semidiscrete method exactly conserves the system energy. Furthermore, we show that with a symplectic Euler time discretization, a perturbed energy quantity exactly conserves perturbed energy that is positive-definite and equivalent to the actual energy under a CFL condition. In addition to proving optimal-order L_infinity (L_2) estimates, we derive stability and error bounds for the time derivatives and divergence of solutions which go beyond the standard estimates in literature.

We apply the Galerkin finite element method to a class of nonlinear Klein-Gordon equations. We give the optimal-order error estimates of Galerkin finite element methods for non-Lipschitz nonlinearity. The result holds in dimensions one and two for general nonlinear term, and in dimension three under a certain growth condition. Our results are the first known for the non-Lipschitz nonlinearity. Time-stepping schemes and numerical results are analyzed to strengthen the theoretical result.
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