Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

A posteriori error estimates for fully discrete fractional-step ϑ-approximations for parabolic equations

This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record A posteriori error estimates for fully discrete fractional-step ϑ-approximations for parabolic equations is available online at:http://imajna.oxfordjournals.org/content/early/2015/07/20/imanum.drv035.abstract?sid=ab7d6b71-cb35-42ed-896f-f009b1fdc99eWe derive optimal order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretisation in time we apply the fractional-step #-scheme and for the discretisation in space the finite element method with finite element spaces that are allowed to change with time

Quadruplets and Quintuplets

A high-order pregnancy is always a challenge not only for the couple but also for the obstetrician, the pediatricians, the midwives, and the whole stuff of an obstetric clinic. The breakthroughs of infertility treatments have made more couples to postpone the birth of their children until they feel professionally and financially safe, many times after the age of 40. The advanced age of the mother puts extra pressure to the clinician for immediate success, leading to a rise of high-order pregnancies until the introduction of regulations and laws in many countries. The cost of a quadruplet and quintuplet pregnancy can be unbearable, not only financially but also psychologically. The management of such a pregnancy is also challenging since its beginning and to the end. Modern techniques and methods can also be difficult to be implemented on a quadruplet of quintuplet pregnancy because of the fear of losing four or five embryos at once. At the same time, the limited number of cases makes it almost impossible for studies to be made and guidelines to be established for most of the cases

Antenatal Sonographic Diagnosis of Pharyngeal Teratoma: Our Experience of a Rare Case with Review of the Literature

Background. Teratomas are the most common tumors. They are usually localized in the sacrococcygeal area, while the pharyngeal localization is very rare. The number of cases of stomatopharyngeal teratomas detected prenatally via sonography is very small. Case Report. We present the case of a 24-year-old primipara at 18 weeks' gestation, that at the routine ultrasound scan, the fetus was found with an echogenic mass, filling the stomatopharyngeal cavity and protruding from the mouth. Other abnormalities were not found. Termination of pregnancy was achieved using misoprostol. A female stillborn fetus with a weight of 250 g and length of 25.5 cm was delivered. The postmortem and pathologic examination confirmed the diagnosis. Conclusion. Pharyngeal teratomas can be diagnosed with the use of ultrasounds in utero facilitating parents' counseling in early time

A system of ODEs for a Perturbation of a Minimal Mass Soliton

We study soliton solutions to a nonlinear Schrodinger equation with a saturated nonlinearity. Such nonlinearities are known to possess minimal mass soliton solutions. We consider a small perturbation of a minimal mass soliton, and identify a system of ODEs similar to those from Comech and Pelinovsky (2003), which model the behavior of the perturbation for short times. We then provide numerical evidence that under this system of ODEs there are two possible dynamical outcomes, which is in accord with the conclusions of Pelinovsky, Afanasjev, and Kivshar (1996). For initial data which supports a soliton structure, a generic initial perturbation oscillates around the stable family of solitons. For initial data which is expected to disperse, the finite dimensional dynamics follow the unstable portion of the soliton curve.Comment: Minor edit