We verify functional a posteriori error estimate for obstacle problem proposed by Repin. Simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution of the obstacle problem can be derived. Quality of a numerical approximation obtained by the finite element method is compared with the exact solution and the error of approximation is bounded from above by a majorant error estimate. The sharpness of the majorant error estimate is discussed.
We verify functional a posteriori error estimates proposed by S. Repin for a class of obstacle problems in two space dimensions. New benchmarks with known analytical solution are constructed based on one dimensional benchmark introduced by P. Harasim and J. Valdman. Numerical approximation of the solution of the obstacle problem is obtained by the finite element method using bilinear elements on a rectangular mesh. Error of the approximation is measured by a functional majorant. The majorant value contains three unknown fields: a gradient field discretized by Raviart-Thomas elements, Lagrange multipliers field discretized by piecewise constant functions and a scalar parameter β. The minimization of the majorant value is realized by an alternate minimization algorithm, whose convergence is discussed. Numerical results validate two estimates, the energy estimate bounding the error of approximation in the energy norm by the difference of energies of discrete and exact solutions and the majorant estimate bounding the difference of energies of discrete and exact solutions by the value of the functional majorant.
Auf Grundlage der Greenwicher photographischen Beobachtungen von Sonnenflecken wird die Anzahl der neu entstandenen Fleckengruppen auf der sichtbaren und unsichtbaren Sonnenhalbkugel und ihre mittlere Lebensdauer im Zeitraum 1874-1950 berechnet, uzw. sowohl für die Gesamtoberfläche der Sonne, als auch gesondert für die nördliche und südliche Halbkugel. Für diese zwei Charakteristiken werden die gleitenden Durchschnitte für 27 Sonnenrotationen und die mittleren Jahreswerte wiedergegeben. Die Ausgangs- und Endwerte dieser Charakteristiken werden tabellarisch dargestellt und der Verlauf der gleitenden Durchschnitte in 27 Rotationen wird graphisch verzeichnet.
Es zeigt sich, dass die elfjährige Periode der Relativzahl durch die elfjährige Periode der Anzahl neu entstandener Fleckengruppen hervorgerufen wird, wobei die Anzahl der der neu entstandenen Fleckengruppen der achtzigjährigen Periode praktisch nicht unterliegt. Dagegen wird die achtzigjährige Periode der Relativzahlen durch die achtzigjährige Periode der mittleren Lebensdauer der Fleckengruppen hervorgerufen, wobei die mittlere Lebensdauer praktisch keinen Zusammenhang mit der elfjährigen Periode hat. Der Autor untersucht eine Reihe anderer Eigenschaften des Verlaufs der Anzahl neu entstandener Gruppen und ihrer mittleren Lebensdauer, sowie die Abhängigkeit dieser Charakteristiken von der heliographischen Breite in den Zeitabschnitten der Maxima und Minima der elfjährigen Zyklen. and Na str. 402-444 tabulky
This study introduces the concept of vernacularization in the context of the literary history ofBohemia around 1800. National philologists, to some extent until today, examine this literaturebased on 19th-century national and aesthetic criteria (i.e. the notion of „genius“, originality etc.)which, as the author argues, do not suit an analysis of multi-lingual pre-Romantic culture. Withoutintending to replace the popular and politically relevant narrative of the National Revival, theconcept of vernacularization attempts to generate a comparatively oriented discussion regardingthe transition (beginning around 1760) from the multi-lingual cultures of a stratified society (thenobility, the clergy, the common people etc.) into separate, linguistically defined regional andsubsequently national cultures and especially national literatures in the first half of the 19th century.Vernacularization is defined as a form of knowledge transfer between cultures considered tohave different places in a European cultural hierarchy. The „higher“ or „classical“ cultures serveas the vehicles for the transfer of culture; they are supposed to be quite independent of regionalcontexts and thus can be interregionally recognized as exemplary; in a stratified society theyare accessible mainly to the elites. That predestines them to serve as a means of representation.„Vernacularization“ indicates the efforts by a region’s intellectual elites to make this arcaneknowledge (or at least its „useful“ parts) accessible to their uneducated compatriots (in theMiddle Ages mainly to the secular elite, in the 18th and 19th centuries above all to the „folk“). Thisdissemination of useful knowledge in support of the general good is described aptly by JosephAnton Riegger as the obligation of the ideal „enlightened patriot.“Therefore, the „logic“ of vernacularization should not be limited to one country or one era; onthe contrary, the concept should encourage comparison and simultaneously provide insightinto the inner hierarchy of European cultures into which regional culture would be integrated.In this context, all „mature“ cultures (not only those of antiquity) can be considered exemplaryor model cultures. The theme of knowledge transfer as a service to the homeland, in spite ofsignificant differences determined by time and place, can be traced through various examples:from Cicero (Greece-R ome) to Dante Alighieri (Roman and Provençal culture to Italy), Du Bellay(Roman and Italian culture to France) and finally to Frederick II (Italian, English and Frenchculture to protestant Germany), through the inaugural lecture (1765) of the Freiburg (and laterPrague) professor of law Joseph Anton Riegger, whose detailed defense of his decision tolecture in German rather than in Latin is a central text in this study. Vernacularization is motivatednot only by a enlightened utilitarian knowledge transfer to serve the own land, but also by thedesire to see one’s own land included in the the hierarchy of „enlightened“ nations.The author also suggests that vernacularization, in the sense of adapting an already establishedhigh culture to a regional level, should not be limited to the medium of language. The role oflanguage is admittedly as important as it is problematic; for example, in the case of the multiethnicBohemian lands, a linguistic doubling took place. This problem is illustrated by the notuncontestedintroduction of German-language lectures at the university in Prague, „identitypolitics“-motivated attempts to establish the equivalence of local languages (F. J. von Kinský’s1773 call, written in German, to regard Czech as the language of the „Czech lands“) or other –Czech and German – defenses of vernacular language(s), literature(s) and culture(s) versusthe established elite cultures such as Latin, French or (north) German. Especially significantare institutions that are either vernacularized by the „enlightened“ higher classes for utilitarianreasons (e.g. originally Latin-based educational institutions, as well as their libraries, which areopened to the public) or institutions that were created for the transfer of knowledge and, usually(co-) established by the secular elite, combine both patriotic motives: compulsory education inthe vernacular language, semi-public associations and institutions such as the Freemasons,reading and lending libraries, newspaper associations, museums (e.g. the „patriotic“ museumof the Prague typographer J. F. Schönfeld) etc. The techniques of vernacularization in the key area of literature receive particular attention. Forexample, in the case of almanacs, which „classic“ examples of poetry are translated, imitatedand adapted? The German-B ohemian and Czech poetical texts draw to some extent fromthe same sources (Latin antiquity, North German anacreontic) but in part from very different„classics“: for Czech literature, a significant role is played by its own classical era of Humanism(around 1600) as a transmitter of the Roman-Latin classics.The author also points out differentiation, from the 1790s on, in the vernacularization of contentand themes. He supports his thesis with examples from the area of museum collections andabove all on the basis of the „vernacularization of myth.“ Here he refers to the idea alreadyexpressed around 1800 by contemporaries (e.g. the Schlegel brothers or Jacob Grimm’s„German Mythology“, 1835) that a national literature without its own mythology after theexample of the Greeks is necessarily incomplete and inferior. In this light, Václav Hanka’s „earlymedieval“ Czech epics (1817ff.) do not appear simply as counterfeits. Instead, they function asthe ultimate act of vernacularization: they give Czech culture a high-cultural mythical beginning.Thus the construction of an “original“ national mythology (after the Greek model) enablesa transition from vernacularization into an autonomous national literature. This national literatureconsequently may draw from all other „classical“ models but no longer has the obligation todo so. It has become a complete, autonomous and „classical“ culture in its own right and hasclaimed a “younger” place in the hierarchy of European (literary) cultures.