The numerical solution of the flow a liquid crystal governed by particular instance Ericksen--Leslie equations is considered. Convergence finite element approximations established under appropriate regularity hypotheses, and experiments exhibiting interaction singularities coupling director momentum are presented.

Article A Delaunay based numerical method for three dimensions: generation, formulation, and partition Share on Authors: Gary L. Miller School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania PennsylvaniaView Profile , Dafna Talmor Shang-Hua Teng Department University Minnesota, Minneapolis, Minnesota MinnesotaView Noel Walkington Mathematics, Authors Info & Claims STOC '95: Proceedings the twenty-seventh annual ACM symposium Theory computingMay 1995 Pages...

Article Smoothing and cleaning up slivers Share on Authors: Herbert Edelsbrunner Department of Computer Science, Duke University, Durham, NC Raindrop Geomagic, Research Triangle Park, NCView Profile , Xiang-Yang Li University Illinois at Urbana-Champaign, Urbana, IL ILView Gary Miller Carnegie Mellon Pittsburgh, PA PAView Andreas Stathopoulos College William Mary, Williamsburg, VA VAView Dafna Talmor LMS-CADSI, Suite 104, 3150 Almaden Expwy, San Jose, CA CAView Shang-Hua Teng Akamai...

A system of quasilinear degenerate parabolic equations arising in the modeling diffusion a fissured medium is studied. There one such equation local cell coordinates at each point medium, and these are coupled through similar global coordinates. It shown that initial boundary value problems well posed appropriate spaces.

The classical discontinuous Galerkin method for a general parabolic equation is analyzed. Symmetric error estimates schemes of arbitrary order are presented. ideas developed below relax many assumptions required in previous work. For example, different discrete spaces may be used at each time step, and the spatial operator need not self-adjoint or independent time. Our posed terms projections exact solution onto valid under minimal regularity guaranteed by natural energy estimate. These...

We consider numerical approximations of incompressible Newtonian fluids having variable, possibly discontinuous, density and viscosity. Since solutions the equations with variable viscosity may not be unique, schemes converge. If solution is then approximate computed using discontinuous Galerkin method to convection stable finite element momentum equation converge solution. a subsequence these will

While Nesterov's algorithm for computing the minimum of a convex function is now over forty years old, it rarely presented in texts first course optimization. This unfortunate since many problems this superior to ubiquitous steepest descent algorithm, and equally simple implement. article presents an elementary analysis that parallels descent. It envisioned presentation could easily be covered few lectures following introductory material on functions included every

Abstract Phase-field fracture models provide a powerful approach to modeling fracture, potentially enabling the unguided prediction of crack growth in complex patterns. To ensure that only tensile stresses and not compressive drive growth, several have been proposed, aiming distinguish between loads. However, these splitting critical shortcoming: they do account for direction hence, are unable crack-normal crack-parallel not. In this study, we apply phase-field model, developed our earlier...

The numerical solution of the flow a liquid crystal governed by particular instance Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve H2(Ω) norms director field. We show how mixed method may be used to eliminate need Hermite finite elements and establish convergence method.

We propose a finite element algorithm for computing the motion of surface moving by mean curvature. The uses level set formulation so that changes in topology can be accommodated. Stability is deduced showing discrete solutions satisfy both $L^\infty $ and $W^{1.1} bounds. Existence connections with Brakke flows are established. Some numerical examples application to related problems, such as phase field equations, also presented.

An algorithm is proposed for the solution of non-convex variational problems. In order to avoid representing highly oscillatory functions on a mesh, an associated Young measure, which characterizes such oscillations, also approximated. Sample calculations demonstrate viability this approach.

Numerical approximation of the flow liquid crystals governed by Ericksen-Leslie equations is considered. Care taken to develop numerical schemes which inherit Hamiltonian structure these and associated stability properties. For a large class material parameters compactness discrete solutions established guarantees convergence.

Discontinuous Galerkin time discretizations are combined with the mixed finite element and continuous methods to solve miscible displacement problem. Stable schemes of arbitrary order in space obtained. Under low regularity assumptions on data, convergence scheme is proved by using compactness results for functions that may be discontinuous time.

We illustrate how some interesting new variational principles can be used for the numerical approximation of solutions to certain (possibly degenerate) parabolic partial differential equations. One remarkable feature algorithms presented here is that derivatives do not enter into principles, so, example, discontinuous approximations may approximating heat equation. present formulae computing a Wasserstein metric which enters formulations.

1. Planting the seeds: introducing maple 2. Numbers, functions and basic algebra 3. Calculus differential equations 4. Matrices, linear package 5. The last resort: numerical methods 6. Graphs graphics 7. Algebra with 8. Useful utilities 9. What's in packages? 10. Looping, branching data structures 11. Introducing programming 12. Programming examples.

A free-boundary problem of Stefan type is presented under constitutive assumptions on flux and energy which contain an effective time delay. This contains the hyperbolic telegraphers equation and, hence, has feature that propagation speed disturbances bounded. With appropriate physically consistent condition interface this shown to lead a well-posed weak formulation problem.

We consider linear first order scalar equations of the form $\rho_t + {\rm div}(\rho v) a \rho = f$ with appropriate initial and boundary conditions. It is shown that approximate solutions computed using discontinuous Galerkin method will converge in $\LtwoLtwo$ when coefficients v data f satisfy minimal assumptions required to establish existence uniqueness solutions. In particular, need not be Lipschitz, so characteristics equation may defined, being approximated my have bounded variation.

Numerical schemes to compute approximate solutions of the evolutionary Stokes and Navier-Stokes equations are studied. The discontinuous in time conforming space arbitrarily high order. Fully-discrete error estimates derived dependence viscosity constant is carefully tracked. It shown that errors bounded by projection exact solution which exhibit optimal rates when smooth.

Abstract The numerical analysis of stochastic parabolic partial differential equations the form $$\begin{aligned} du + A(u)\, dt = f \,dt g \, dW, \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mi>d</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mo>)</mml:mo> <mml:mspace/> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mi>g</mml:mi> <mml:mi>W</mml:mi>...