Gradient schemes for the Stefan problem

We show in this paper that the gradient schemes (which encompass a large family of discrete schemes) may be used for the approximation of the Stefan problem $\partial_t \bar u - \Delta \zeta (\bar u) = f$. The convergence of the gradient schemes to the continuous solution of the problem is proved thanks to the following steps. First, estimates show (up to a subsequence) the weak convergence to some function $u$ of the discrete function approximating $\bar u$. Then Alt-Luckhaus' method, relying on the study of the translations with respect to time of the discrete solutions, is used to prove that the discrete function approximating $\zeta(\bar u)$ is strongly convergent (up to a subsequence) to some continuous function $\chi$. Thanks to Minty's trick, we show that $\chi = \zeta(u)$. A convergence study then shows that $u$ is then a weak solution of the problem, and a uniqueness result, given here for fitting with the precise hypothesis on the geometric domain, enables to conclude that $u = \bar u$. This convergence result is illustrated by some numerical examples using the Vertex Approximate Gradient scheme

Teoremi ulaganja Soboljevljevih prostora i primjene

U ovom radu dokazali smo teoreme ulaganja Soboljevljevih prostora $W^{k,q}$ u $L^p$ prostore (ili čak Hölderove prostore, ako je $kq > n$, gdje je $n$ dimenzija prostora $\mathbb{R}^n$) na ograničenim (dovoljno glatkim) domenama. Među njima je posebno važan Rellich-Kondrachovljev teorem, koji osigurava kompaktnost određenih ulaganja, što smo u dva primjera iskoristili za dokaz egzistencije rješenja nelinearnih rubnih zadaća. Prvi je primjer kvazilinearna eliptička PDJ s homogenim rubim uvjetom \begin{align*} \begin{cases} -\Delta u + b(Du) + \mu u =0 &\text{na } \Omega\\ u = 0 & \text{na } \partial \Omega, \end{cases} \end{align*} gdje je $\Omega$ ograničena domena klase $C^2$. Pretpostavili smo da je $b : \mathbb{R}^n \to \mathbb{R}$ Lipschitzova te zadovoljava uvjet na rast $\left| b(p) \right| \leq C(\left| p \right| + 1)$ za neki $C$ i za sve $p \in \mathbb{R}^n$. Egzistencija rješenja $u \in H^2(\Omega) \cap H_0^1(\Omega$) ove zadaće pokazana je pomoću Schauderovog teorema o fiksnoj točki, čija je pretpostavka kompaktnosti odgovarajućeg operatora osigurana kompaktnošću ulaganja $H^2(\Omega) \subset H^1(\Omega)$. Drugi je primjer stacionarna Navier-Stokesova zadaća, također s homogenim rubnim uvjetom, na ograničenoj glatkoj domeni $\Omega$, \begin{align*} \begin{cases} -\mu \Delta \textbf{u} + (D \textbf{u})\textbf{u} + \text{grad } p = \textbf{f} &\text{na } \Omega\\ div \: \textbf{u} = 0 &\text{na } \Omega\\ \textbf{u} = 0 &\text{na } \Gamma, \end{cases} \end{align*} gdje su vektorska funkcija $\textbf{u} = (u_1, \dots , u_n)$ i skalarna funkcija $p$ tražene funkcije, dok je $\textbf{f}$ zadana. Ovdje je egzistencija rješenja pokazana pomoću Galerkinove metode, dakle konstruirali smo ograničen niz aproksimativnih rješenja te na limesu nekog njegovog podniza dobili rješenje. Za to nam je bila potrebna kompaktnost ulaganja $H_0^1(\Omega) \subset L^2(\Omega)$.In this thesis we proved the Sobolev embedding theorems of spaces $W^{k,q}(\Omega)$ in $L^p$ spaces (or even Hölder spaces, when $kq > n$, where $n$ is the dimension of $\mathbb{R}^n$) on bounded (smooth enough) domains. Among them, the Rellich-Kondrachov Theorem is particularly important, since it provides compactness of certain embeddings, which we have used in two examples to prove the existence of solutions of nonlinear boundary problems. The first example is the quasilinear elliptic PDE with homogeneous boundary condition \begin{align*} \begin{cases} -\Delta u + b(Du) + \mu u =0 &\text{on } \Omega\\ u = 0 & \text{on } \partial \Omega, \end{cases} \end{align*} where $\Omega$ is a bounded domain of class $C^2$. We assumed that $b : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz continous and satisfies the growth condition $\left| b(p) \right| \leq C(\left| p \right| + 1)$ for some $C$ and all $p \in \mathbb{R}^n$. The existence of a solution $u \in H^2(\Omega) \cap H_0^1(\Omega)$ of this problem is shown using Schauder’s Fixed Point Theorem, whose assumption of compactness of the corresponding operator is secured by the compactness of the embedding $H^2(\Omega) \subset H^1(\Omega)$. The second example is the stationary Navier-Stokes problem, also with homogeneous boundary condition, on a bounded smooth domain $\Omega$, \begin{align*} \begin{cases} -\mu \Delta \textbf{u} + (D \textbf{u})\textbf{u} + \text{grad } p = \textbf{f} &\text{on } \Omega\\ div \: \textbf{u} = 0 &\text{on } \Omega\\ \textbf{u} = 0 &\text{on } \Gamma, \end{cases} \end{align*} where the vector function $\textbf{u} = (u_1, \dots , u_n)$ and the scalar function $p$ are required functions, while $\textbf{f}$ is given. Here the existence of a solution is shown using the Galerkin method, i.e. we have constructed a bounded sequence of approximate solutions and got the solution as the limit of its subsequence. To do this, we needed the compactness of the embedding $H_0^1(\Omega) \subset L^2(\Omega)$

Teoremi ulaganja Soboljevljevih prostora i primjene

U ovom radu dokazali smo teoreme ulaganja Soboljevljevih prostora $W^{k,q}$ u $L^p$ prostore (ili čak Hölderove prostore, ako je $kq > n$, gdje je $n$ dimenzija prostora $\mathbb{R}^n$) na ograničenim (dovoljno glatkim) domenama. Među njima je posebno važan Rellich-Kondrachovljev teorem, koji osigurava kompaktnost određenih ulaganja, što smo u dva primjera iskoristili za dokaz egzistencije rješenja nelinearnih rubnih zadaća. Prvi je primjer kvazilinearna eliptička PDJ s homogenim rubim uvjetom \begin{align*} \begin{cases} -\Delta u + b(Du) + \mu u =0 &\text{na } \Omega\\ u = 0 & \text{na } \partial \Omega, \end{cases} \end{align*} gdje je $\Omega$ ograničena domena klase $C^2$. Pretpostavili smo da je $b : \mathbb{R}^n \to \mathbb{R}$ Lipschitzova te zadovoljava uvjet na rast $\left| b(p) \right| \leq C(\left| p \right| + 1)$ za neki $C$ i za sve $p \in \mathbb{R}^n$. Egzistencija rješenja $u \in H^2(\Omega) \cap H_0^1(\Omega$) ove zadaće pokazana je pomoću Schauderovog teorema o fiksnoj točki, čija je pretpostavka kompaktnosti odgovarajućeg operatora osigurana kompaktnošću ulaganja $H^2(\Omega) \subset H^1(\Omega)$. Drugi je primjer stacionarna Navier-Stokesova zadaća, također s homogenim rubnim uvjetom, na ograničenoj glatkoj domeni $\Omega$, \begin{align*} \begin{cases} -\mu \Delta \textbf{u} + (D \textbf{u})\textbf{u} + \text{grad } p = \textbf{f} &\text{na } \Omega\\ div \: \textbf{u} = 0 &\text{na } \Omega\\ \textbf{u} = 0 &\text{na } \Gamma, \end{cases} \end{align*} gdje su vektorska funkcija $\textbf{u} = (u_1, \dots , u_n)$ i skalarna funkcija $p$ tražene funkcije, dok je $\textbf{f}$ zadana. Ovdje je egzistencija rješenja pokazana pomoću Galerkinove metode, dakle konstruirali smo ograničen niz aproksimativnih rješenja te na limesu nekog njegovog podniza dobili rješenje. Za to nam je bila potrebna kompaktnost ulaganja $H_0^1(\Omega) \subset L^2(\Omega)$.In this thesis we proved the Sobolev embedding theorems of spaces $W^{k,q}(\Omega)$ in $L^p$ spaces (or even Hölder spaces, when $kq > n$, where $n$ is the dimension of $\mathbb{R}^n$) on bounded (smooth enough) domains. Among them, the Rellich-Kondrachov Theorem is particularly important, since it provides compactness of certain embeddings, which we have used in two examples to prove the existence of solutions of nonlinear boundary problems. The first example is the quasilinear elliptic PDE with homogeneous boundary condition \begin{align*} \begin{cases} -\Delta u + b(Du) + \mu u =0 &\text{on } \Omega\\ u = 0 & \text{on } \partial \Omega, \end{cases} \end{align*} where $\Omega$ is a bounded domain of class $C^2$. We assumed that $b : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz continous and satisfies the growth condition $\left| b(p) \right| \leq C(\left| p \right| + 1)$ for some $C$ and all $p \in \mathbb{R}^n$. The existence of a solution $u \in H^2(\Omega) \cap H_0^1(\Omega)$ of this problem is shown using Schauder’s Fixed Point Theorem, whose assumption of compactness of the corresponding operator is secured by the compactness of the embedding $H^2(\Omega) \subset H^1(\Omega)$. The second example is the stationary Navier-Stokes problem, also with homogeneous boundary condition, on a bounded smooth domain $\Omega$, \begin{align*} \begin{cases} -\mu \Delta \textbf{u} + (D \textbf{u})\textbf{u} + \text{grad } p = \textbf{f} &\text{on } \Omega\\ div \: \textbf{u} = 0 &\text{on } \Omega\\ \textbf{u} = 0 &\text{on } \Gamma, \end{cases} \end{align*} where the vector function $\textbf{u} = (u_1, \dots , u_n)$ and the scalar function $p$ are required functions, while $\textbf{f}$ is given. Here the existence of a solution is shown using the Galerkin method, i.e. we have constructed a bounded sequence of approximate solutions and got the solution as the limit of its subsequence. To do this, we needed the compactness of the embedding $H_0^1(\Omega) \subset L^2(\Omega)$

Teoremi ulaganja Soboljevljevih prostora i primjene

U ovom radu dokazali smo teoreme ulaganja Soboljevljevih prostora $W^{k,q}$ u $L^p$ prostore (ili čak Hölderove prostore, ako je $kq > n$, gdje je $n$ dimenzija prostora $\mathbb{R}^n$) na ograničenim (dovoljno glatkim) domenama. Među njima je posebno važan Rellich-Kondrachovljev teorem, koji osigurava kompaktnost određenih ulaganja, što smo u dva primjera iskoristili za dokaz egzistencije rješenja nelinearnih rubnih zadaća. Prvi je primjer kvazilinearna eliptička PDJ s homogenim rubim uvjetom \begin{align*} \begin{cases} -\Delta u + b(Du) + \mu u =0 &\text{na } \Omega\\ u = 0 & \text{na } \partial \Omega, \end{cases} \end{align*} gdje je $\Omega$ ograničena domena klase $C^2$. Pretpostavili smo da je $b : \mathbb{R}^n \to \mathbb{R}$ Lipschitzova te zadovoljava uvjet na rast $\left| b(p) \right| \leq C(\left| p \right| + 1)$ za neki $C$ i za sve $p \in \mathbb{R}^n$. Egzistencija rješenja $u \in H^2(\Omega) \cap H_0^1(\Omega$) ove zadaće pokazana je pomoću Schauderovog teorema o fiksnoj točki, čija je pretpostavka kompaktnosti odgovarajućeg operatora osigurana kompaktnošću ulaganja $H^2(\Omega) \subset H^1(\Omega)$. Drugi je primjer stacionarna Navier-Stokesova zadaća, također s homogenim rubnim uvjetom, na ograničenoj glatkoj domeni $\Omega$, \begin{align*} \begin{cases} -\mu \Delta \textbf{u} + (D \textbf{u})\textbf{u} + \text{grad } p = \textbf{f} &\text{na } \Omega\\ div \: \textbf{u} = 0 &\text{na } \Omega\\ \textbf{u} = 0 &\text{na } \Gamma, \end{cases} \end{align*} gdje su vektorska funkcija $\textbf{u} = (u_1, \dots , u_n)$ i skalarna funkcija $p$ tražene funkcije, dok je $\textbf{f}$ zadana. Ovdje je egzistencija rješenja pokazana pomoću Galerkinove metode, dakle konstruirali smo ograničen niz aproksimativnih rješenja te na limesu nekog njegovog podniza dobili rješenje. Za to nam je bila potrebna kompaktnost ulaganja $H_0^1(\Omega) \subset L^2(\Omega)$.In this thesis we proved the Sobolev embedding theorems of spaces $W^{k,q}(\Omega)$ in $L^p$ spaces (or even Hölder spaces, when $kq > n$, where $n$ is the dimension of $\mathbb{R}^n$) on bounded (smooth enough) domains. Among them, the Rellich-Kondrachov Theorem is particularly important, since it provides compactness of certain embeddings, which we have used in two examples to prove the existence of solutions of nonlinear boundary problems. The first example is the quasilinear elliptic PDE with homogeneous boundary condition \begin{align*} \begin{cases} -\Delta u + b(Du) + \mu u =0 &\text{on } \Omega\\ u = 0 & \text{on } \partial \Omega, \end{cases} \end{align*} where $\Omega$ is a bounded domain of class $C^2$. We assumed that $b : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz continous and satisfies the growth condition $\left| b(p) \right| \leq C(\left| p \right| + 1)$ for some $C$ and all $p \in \mathbb{R}^n$. The existence of a solution $u \in H^2(\Omega) \cap H_0^1(\Omega)$ of this problem is shown using Schauder’s Fixed Point Theorem, whose assumption of compactness of the corresponding operator is secured by the compactness of the embedding $H^2(\Omega) \subset H^1(\Omega)$. The second example is the stationary Navier-Stokes problem, also with homogeneous boundary condition, on a bounded smooth domain $\Omega$, \begin{align*} \begin{cases} -\mu \Delta \textbf{u} + (D \textbf{u})\textbf{u} + \text{grad } p = \textbf{f} &\text{on } \Omega\\ div \: \textbf{u} = 0 &\text{on } \Omega\\ \textbf{u} = 0 &\text{on } \Gamma, \end{cases} \end{align*} where the vector function $\textbf{u} = (u_1, \dots , u_n)$ and the scalar function $p$ are required functions, while $\textbf{f}$ is given. Here the existence of a solution is shown using the Galerkin method, i.e. we have constructed a bounded sequence of approximate solutions and got the solution as the limit of its subsequence. To do this, we needed the compactness of the embedding $H_0^1(\Omega) \subset L^2(\Omega)$

Closed form asymptotics for local volatility models

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods.Comment: 30 pages, 10 figure

Influence of post-cyclic loading on hemic peat

Construction on peat soils has proven to be a challenging task to civil engineers since this soil type has a significant issue that arises from common problems construction of roads, housing and embankment construction with regard to peat are stability, settlements and major problems were encountered especially on deep peat. For many years, in road design as an example, static loading method was applied in road designed by considering soil shear strength through static load and do not take into account the vehicular dynamic loading and shear strength thereafter. This fact is related to the shear strength of peat soil after dynamically loaded. The aim of this research is to establish the post-cyclic behaviour of peat soil after cyclically loaded and to assess the effect of parameters changes on static and post-cyclic behaviour of peat soil. 200 specimens are tested, and prepared under consolidated undrained triaxial with effective stresses at 25kPa, 50 kPa, and 100 kPa with different location from Parit Nipah, Johor, Parit Sulong, Batu Pahat, Johor and Beaufort, Sabah. These specimens tested using GDS Enterprise Level Dynamic Triaxial Testing System (ELDYN) apparatus. Whereas, dynamic load tests are carried out in different frequencies to simulate the loading type such as vibration of machineries, wind, traffic load and earthquake in field from 1.0 Hz, 2.0 Hz and 3.0 Hz with 100 numbers of loading cycles. Post-cyclic monotonic shear strength results and then compared to the static monotonic results. Significantly, showed some vital changes that leads to the changes of stress-strain behaviour. Apparently, the result shows that post-cyclic shear strength decreases with the increase of frequencies. Prior to critical yield strain level, the peat specimen experience a significant deformation. The deformation of peats triggers changes in soil structures that causes reduction in stress-strain behaviour. Thus, it can be concluded that the stress-strain behaviour of peat soil decreased after 100 numbers of cyclic loading in post-cyclic test as compared to the static tests, and it decreased substantially when frequencies were applied. The post-cyclic specimen had a lower undrained parameters than did the static. Reduction of cohesion value in postcylic compared to static almost 70% and reduction of friction angle is about 46.34%