<p>We are made of stardust in the sense that most atomic nuclei around us have been formed by stars. Stars synthesise new elements and expel them to the interstellar medium, from which later new generations of stars are born. We can map this chemical evolution by analysing the atmospheric contents of old Galactic halo stars. I have done two such investigations. A vigourous debate is going on whether the oxygen-to-iron ratio varies strongly with the general metal-content of halo stars. In my first study, I made an abundance analysis of 43 halo stars, and found no support for such a variation. I have also found that there probably is a cosmic spread in the abundances of oxygen, magnesium, silicon, and calcium relative to iron for halo stars. This may be an indication that the halo was built up by subsystems with differences in the star formation rate. In my second study, I performed a thorough abundance analysis of the star HE0338-3945, which is strangely overabundant in both r- and s-elements. Several other stars have been found with abundance patterns curiously similar to this star, and I define new criteria for the class r+s stars. The abundance similarities among the r+s stars suggest a common formation scenario. However, as the s-elements usually are considered to be produced in binary systems of low mass, and r-elements in supernovae of Type II, this scenario is not obvious. In the article I discuss seven hypotheses, and several of them are dismissed.</p>

We are made of stardust in the sense that most atomic nuclei around us have been formed by stars. Stars synthesise new elements and expel them to the interstellar medium, from which later new generations of stars are born. We can map this chemical evolution by analysing the atmospheric contents of old Galactic halo stars. I have done two such investigations. A vigourous debate is going on whether the oxygen-to-iron ratio varies strongly with the general metal-content of halo stars. In my first study, I made an abundance analysis of 43 halo stars, and found no support for such a variation. I have also found that there probably is a cosmic spread in the abundances of oxygen, magnesium, silicon, and calcium relative to iron for halo stars. This may be an indication that the halo was built up by subsystems with differences in the star formation rate. In my second study, I performed a thorough abundance analysis of the star HE0338-3945, which is strangely overabundant in both r- and s-elements. Several other stars have been found with abundance patterns curiously similar to this star, and I define new criteria for the class r+s stars. The abundance similarities among the r+s stars suggest a common formation scenario. However, as the s-elements usually are considered to be produced in binary systems of low mass, and r-elements in supernovae of Type II, this scenario is not obvious. In the article I discuss seven hypotheses, and several of them are dismissed.

In the dissertation we discuss the rectangular elements, Adini's elements and $p-$order Lagrange elements, which were constructed in the rectangular finite spaces. The special rectangular partitions enable the finite element solutions $u_h$ more efficient in interpolation of the true solution for Elliptic equation $u_I$. The convergence rates of $|u_h-u_I|_1$ are one or two orders higher than the optimal convergence rates. For post-processings we construct higher order interpolation operation $Pi_p$ to reach superconvergence $|u-Pi_p u_h|_1$. To our best knowledge, we at the first time provided the a posteriori interpolant formulas of Adini's elements and biquadratic Lagrange elements to obtain the global superconvergence, and at the first time reported the numerical verification for supercloseness $O(h^4)-O(h^5) $, global superconvergence $O(h^5)$ in $H^1$-norm and the high rates $O(h^6|ln h|)$ in the infinity norm for Poisson's equation(i.e., $-Delta u = f$). Since the finite element methods is fail to deal with the singularity problems, in the dissertation, the combinations of the Ritz-Galerkin method and the finite element methods are used for the singularity problem, i.e., Motz's problem. To couple two methods along their common boundary, we adopt the simplified hybrid, penalty, and penalty plus hybrid techniques. The analysis are made in the dissertation to derive the almost best global superconvergence $O(h^{p+2-delta})$ in $H^1$-norm, $0<delta << 1$, for the combination using $p(geq 2)$-rectangles in the smooth subdomain, and the best global superconvergence $O(h^{3.5})$ in $H^1$-norm for combinations of Adini's elements in the smooth subdomain. The numerical experiments have been carried out for the combinations of the Ritz-Galerkin method and Adini's elements, to verify the theoretical superconvergence derived.