The dissertation consists of two parts.The first part is mainly to provide the algorithms and error estimates of the collocation Trefftz methods (CTMs) for seeking the solutions of partial differential equations. We consider several popular models of PDEs with singularities, including Poisson equations and the biharmonic equations. The second part is to present the collocation methods (CMs) and to give a unified framework of combinations of CMs with other numerical methods such as finite element method, etc. An interesting fact has been justified: The integration quadrature formulas only affect on the uniformly $V_h$-elliptic inequality, not on the solution accuracy. In CTMs and CMs, the Gaussian quadrature points will be chosen as the collocation points. Of course, the Newton-Cotes quadrature points can be applied as well. We need a suitable dense points to guarantee the uniformly $V_h$-elliptic inequality. In addition, the solution domain of problems may not be confined in polygons. We may also divide the domain into several small subdomains. For the smooth solutions of problems, the different degree polynomials can be chosen to approximate the solutions properly. However, different kinds of admissible functions may also be used in the methods given in this dissertation. Besides, a new unified framework of combinations of CMs with other methods will be analyzed. In this dissertation, the new analysis is more flexible towards the practical problems and is easy to fit into rather arbitrary domains. Thus is a great distinctive feature from that in the existing literatures of CTMs and CMs. Finally, a few numerical experiments for smooth and singularity problems are provided to display effectiveness of the methods proposed, and to support the analysis made.

We study a new method for the strong approximate solution of stochastic differential equations using coupling and we prove order one error bounds for the new scheme in Lp space assuming the invertibility of the diffusion matrix. We introduce and implement two couplings called the exact and approximate coupling for this scheme obtaining good agreement with the theoretical bound. Also we describe a method for non-invertibility case (Combined method) and we investigate its convergence order which will give O(h3/4 √log(h)j) under some conditions. Moreover we compare the computational results for the combined method with its theoretical error bound and we have obtained a good agreement between them. In the last part of this thesis we work out the performance of the multilevel Monte Carlo method using the new scheme with the exact coupling and we compare the results with the trivial coupling for the same scheme.

The singular solutions for linear elastostatics at corners are essential in both theory and computation. In this thesis, we seek new singular solutions for corners with the fixed (displacement), the free stress (traction) boundary conditions, and their mixed types, and to explore their corner singularity and provide the algorithms and error estimates in detail. The singular solutions of linear elastostatics are derived, and a number of new models of corner and crack singularity are proposed. Effective numerical methods, such as the collocation Trefftz methods (CTM), the method of fundamental solutions (MFS), the method of particular solutions (MPS) and their combinations: the so called combined method, are developed. Such solutions are useful to examine other numerical methods for singularity problems in linear elastostatics. This thesis consists of three parts, Part I: Basic approaches, Part II: Advanced topics, and Part III: Mixed types of displacement and traction conditions. Contents of Parts I and II have been published in [47,82]. In Part I, the collocation Trefftz methods are used to obtain highly accurate solutions, where the leading coefficient has 14 (or 13) significant digits by the computation with double precision. In part II, two more new models (symmetric and anti-symmetric) of interior crack singularities are proposed, for the corner and crack singularity problems, the combined methods by using many fundamental solutions, but by adding a few singular solutions are proposed. Such a kind of combined methods is significant for linear elastostatics with corners (i.e., the L-shaped domain), because the singular solutions can only be obtained by seeking the power £hk of r£hk numerically. Hence, only a few singular solutions used may greatly simplify the numerical algorithms; Part III is a continued study of Parts I and II, to explore mixed type of displacement and free traction boundary conditions. To our best knowledge, this is the first time to provide the particular solutions near the corner with mixed types of boundary conditions and to report their numerical computation with different boundary conditions on the same corner edge in linear elastostatics. This thesis explores corner singularity and its numerical methods, to form a systematic study of basic theory and advanced computation for linear elastostatics.

LIBs have become an ideal choice in the EVs batteries and stationary storage applications due to their invaluable advantages. The demand for LIBs is projected to increase due to the growth in the sale of EVs. But an average lifespan of LIBs is expected to be around 8 years, thus resulting in the generation of a huge pile of spent batteries after their end of life. The disposal of spent LIBs has several environmental impacts and also results in the loss of valuable metals as a waste. So, efficient, and sustainable recycling methods should also grow at the same pace as of the EVs industries. Li is one of the vital elements for the modern energy revolution and there is a growing demand for its usage in the battery applications. High demand for Li along with supply risks due to its uneven distribution in different geographical locations will increase the Li price. Also, Li was added to the list of EUs critical materials. Currently, only 1% of Li from the end of life products is being recovered. The hydrometallurgy processes based on acid leaching followed by recovery steps are found to be efficient in the recovery of a considerable amount of Li yet there are many drawbacks. Therefore, a combined method such as a thermal treatment followed by water leaching is found to be a promising route to first selectively separate Li from the other valuable metals. The purpose of this research is to investigate the selective leaching of Li from thermally pretreated waste LIBs (by pyrolysis and incineration processes between 400 – 700 ºC for 30, 60, 90min) with water as a leaching agent at high temperature and low L/S ratio. Al salts were also leached along with Li with an efficiency not higher than 3.5%. The finding showed that the time of thermal pretreatment did not have a significant change in Li leaching efficiency. The effect of the S/L ratio showed that the leaching efficiency of Li was higher with an increase in liquid content, keeping the mass of solid constant. At a higher leaching temperature, the leaching efficiency of Li was higher due to an increase in the solubility of Li salts. The highest Li leaching efficiency of nearly 60% was observed from the sample pyrolyzed at 700 ºC for 60 min with the leaching condition S/L ratio = 1:20 g/mL, 80 ºC, 300 rpm, 3 hrs. Furthermore, the tests such as addition of 10% excess carbon during thermal treatment or use of carbonated water during leaching were carried out to enhance the leaching efficiency of Li. However, the test results did not have a significant increase in the leaching efficiency of Li. / LIB har blivit ett idealiskt val i EV-batterier och stationära lagringsapplikationer på grund av deras ovärderliga fördelar. Efterfrågan på LIB bedöms öka på grund av tillväxten i försäljningen av elbilar. Men en genomsnittlig livslängd på LIB förväntas vara cirka 8 år, vilket resulterar i att en enorm hög med förbrukade batterier genereras efter deras livslängd. Bortskaffande av förbrukade LIB har flera miljöpåverkan och resulterar också i förlust av värdefulla metaller som avfall. Så effektiva och hållbara återvinningsmetoder borde också växa i samma takt som inom elbranschen. Li är ett av de viktigaste elementen för den moderna energirevolutionen och det finns en växande efterfrågan på dess användning i batteritillämpningarna. Hög efterfrågan på Li tillsammans med utbudsrisker på grund av dess ojämna fördelning på olika geografiska platser kommer att öka Li-priset. Li läggs också till i listan över EU: s kritiska material. För närvarande återvinns endast 1% av Li från uttjänta produkter. Hydrometallurgiprocesserna baserade på syraläckning följt av återvinningssteg har visat sig vara effektiva vid utvinningen av en betydande mängd Li men det finns många nackdelar. Därför har en kombinerad metod, såsom en termisk behandling följt av vattenlakning, visat sig vara en lovande väg att först selektivt separera Li från de andra värdefulla metallerna. Syftet med denna forskning är att undersöka den selektiva urlakningen av Li från termiskt förbehandlade avfall LIB (genom pyrolys och förbränningsprocesser mellan 400 - 700 ºC under 30, 60, 90 minuter) med vatten som läckmedel vid hög temperatur och låg L / S förhållande. Al-salter lakades också ut tillsammans med Li med en verkningsgrad som inte var högre än 3,5%. Resultatet visade att tiden för termisk förbehandling inte hade någon signifikant förändring av Li-utlakningseffektiviteten. Effekten av S / L-förhållandet visade att urlakningseffektiviteten för Li var högre med en ökning av vätskeinnehållet, vilket höll massan av fast ämne konstant. Vid en högre urlakningstemperatur var urlakningseffektiviteten för Li högre på grund av en ökning av lösligheten av Li-salter. Den högsta Li-lakningseffektiviteten på nästan 60% observerades från provet pyrolyserat vid 700 ° C under 60 minuter med läckningsförhållandet S / L-förhållande = 1:20 g / ml, 80 ° C, 300 rpm, 3 timmar. Vidare utfördes testerna såsom tillsats av 10% överskott av kol under termisk behandling eller användning av kolsyrat vatten under lakning för att förbättra lakningseffektiviteten för Li. Emellertid hade testresultaten ingen signifikant ökning av utlakningseffektiviteten för Li.

Summaries in Afrikaans and English / Die verwerwing van lees- en skryfvaardighede is van wesenlike belang vir die optimale verwerkliking van die mens se potensiele moontlikhede. 'n Ondersoek na veranderende opvattinge oar die aanvangsonderrig van lees en skryf is in die Jig daarvan van belang. Faktore wat aanleiding gegee het tot veranderende opvattinge oar die aanvangsonderrig van lees en skryf kan kortliks saamgevat word as: die ontwikkeling van 'n fonetiese alfabet, veranderende opvattinge oar effektiewe lees- en skryfonderrigmetodes, asook politieke, ekonomiese en samelewingsveranderinge. Hierdie faktore het tot die verbesondering van die Sumeriese, antieke Griekse, antieke Romeinse, Nederlandse, Engelse en Suid-Arikaanse opvoedingspraktyke bygedra. Verskeie onderrigmetodes, onder meer die alfabetmetode, klankmetode, globale metode, gekombineerde metode en eklektiese metode is deur eeue heen in hierdie formele opvoedingspraktyke gebruik vir die aanvangsonderrig van lees en skryf. Effektiewe aanvangslees- en skryfonderrig bepaal of die leerder sy taalvaardighede kan benut om 'n positiewe bydrae tot die samelewing te lewer. / The acquisition of reading and writing skills is of cardinal importance for the optimum realization of man's potential. Research into the changing perspectives which have led to the development of beginning-teaching of reading and writing is therefore of great importance. The following factors have led to changing perspectives: the development of an alphabet, changing perspectives on the effective teaching of reading and writing as well as political, economic and societal changes. These factors contributed to the specific development of the Sumerian, ancient Greek, ancient Roman, Dutch, English and South African education practices. Various teaching methods inter alia the alphabet, phonic, global, combined and eclectic methods have been used in the aforementioned education practices for the beginning-teaching of reading and writing. Effective beginning teaching of reading and writing determines whether the learner will be able to utilize his language skills in order to make a positive contribution to society. / Educational Studies / M. Ed. (Historiese Opvoedkunde)

Summaries in Afrikaans and English / Die verwerwing van lees- en skryfvaardighede is van wesenlike belang vir die optimale verwerkliking van die mens se potensiele moontlikhede. 'n Ondersoek na veranderende opvattinge oar die aanvangsonderrig van lees en skryf is in die Jig daarvan van belang. Faktore wat aanleiding gegee het tot veranderende opvattinge oar die aanvangsonderrig van lees en skryf kan kortliks saamgevat word as: die ontwikkeling van 'n fonetiese alfabet, veranderende opvattinge oar effektiewe lees- en skryfonderrigmetodes, asook politieke, ekonomiese en samelewingsveranderinge. Hierdie faktore het tot die verbesondering van die Sumeriese, antieke Griekse, antieke Romeinse, Nederlandse, Engelse en Suid-Arikaanse opvoedingspraktyke bygedra. Verskeie onderrigmetodes, onder meer die alfabetmetode, klankmetode, globale metode, gekombineerde metode en eklektiese metode is deur eeue heen in hierdie formele opvoedingspraktyke gebruik vir die aanvangsonderrig van lees en skryf. Effektiewe aanvangslees- en skryfonderrig bepaal of die leerder sy taalvaardighede kan benut om 'n positiewe bydrae tot die samelewing te lewer. / The acquisition of reading and writing skills is of cardinal importance for the optimum realization of man's potential. Research into the changing perspectives which have led to the development of beginning-teaching of reading and writing is therefore of great importance. The following factors have led to changing perspectives: the development of an alphabet, changing perspectives on the effective teaching of reading and writing as well as political, economic and societal changes. These factors contributed to the specific development of the Sumerian, ancient Greek, ancient Roman, Dutch, English and South African education practices. Various teaching methods inter alia the alphabet, phonic, global, combined and eclectic methods have been used in the aforementioned education practices for the beginning-teaching of reading and writing. Effective beginning teaching of reading and writing determines whether the learner will be able to utilize his language skills in order to make a positive contribution to society. / Educational Studies / M. Ed. (Historiese Opvoedkunde)

This thesis deals with the topic of partial differential equations parallel solutions. First, it focuses on ordinary differential equations (ODE) and their solution methods using Taylor polynomial. Another part is devoted to partial differential equations (PDE). There are several types of PDE, there are parabolic, hyperbolic and eliptic PDE. There is also explained how to use TKSL system for PDE computing. Another part focuses on solution methods of PDE, these methods are forward, backward and combined methods. There was explained, how to solve these methods in TKSL and Matlab systems. Computing accuracy and time complexity are also discussed. Another part of thesis is PDE parallel solutions. Thanks to the possibility of PDE convertion to ODE systems it is possible to represent each ODE equation by independent operation unit. These units enable parallel computing. The last chapter is devoted to implementation. Application enables generation of ODE systems for TKSL system. These ODE systems represent given hyperbolic PDE.

Vojtěch Stejskal - Diplomová práce 2014 - Přírodovědecká fakulta UK v Praze 4 ABSTRACT The thesis describes pilot applications of combined method - combination of sodium lactate and nanoparticles of zero-valent iron and their synergic effect. Two applications of combined method are described onto two geologically different sites - in Rožmitál pod Třemšínem and Spolchemie in Ústí nad Labem. First site is contaminated by polychlorinated biphenyls; contamination is situated in hydraulically well conducted porous media formed by weathered granodiorite. Main contaminants of Spolchemie site are trichloroethylene and tetrachloroethylene, contamination is situated in geologically heterogenous quaternary terrace of Bílina river. Effects of combined method on two different sites were compared and recommendations and conclusions were done. Part of the thesis is research of scientific articles and literature on topics: polychlorinated biphenyls, chlorinated ethylenes, in situ chemical reduction by sodium lactate and nanoiron, natural conditions of both sites, history of sites. In the thesis are also processed and evaluated results of geophysical investigation, changes of groundwater level and results of groundwater monitoring in view of the application of combined method on both sites.