Partially Penalized Immersed Finite Element Methods for Elliptic Interface Problems

This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only these IFE methods can be proven to have the optimal convergence rate in the H1-norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the H1-norm and the L2-norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods, but also are of a great potential to be useful in error analysis for other IFE methods

The Quark Dirac Sea and the Contracted Universe cooperate to produce the Big Bang

The Big Bang theory cannot and does not provide any explanation for the primordial hot and dense initial condition. In order to give an explanation for the cause of the Big Bang, this paper expands the original Dirac sea (which includes only electrons) to the quark Dirac sea (QDS) including quarks (u and d) for producing the Big Bang with quark energy. The QDS is composed of "relatively infinite" u-quarks and d-quarks as well as electrons with negative energy in the vacuum. A huge number of domains with sizes much smaller than $10^{-18}$m of the body-central cubic quark lattice with a lattice constant "a" = Planck length ($1.62\times10^{-35}m$) are distributed randomly over the QDS. The QDS is a homogeneous, isotropic, equivalent "continuous" and "empty" (no net electric charge, no net color charge, no gravitational force field since the gravitational potential is the same at any physical point in the QDS) perfect vacuum model. The gravity of the universe pulls on the quarks inside the QDS. The pulling force becomes larger and larger as the universe shrinks and shrinks. Once the pulling force is larger than the binding force on the quarks by the whole QDS, a huge number of quarks and antiquarks will be excited out from the QDS. This is a necessary and sufficient condition for the Big Bang. The huge number of excited quark-antiquark pairs annihilate back to the QDS and release a huge amount of energy; these energies make the big bang