PDE Problems Research Articles

Construction of tetrahedral meshes of degree two

SUMMARYThere is a need for finite elements of degree two or more to solve various PDE problems. This paper discusses a method to construct such meshes in the case of tetrahedral element of degree two. The first section of this paper returns to Bézier curves, Bézier triangles and then Bézier tetrahedra of degree two. The way in which a Bézier tetrahedron and a P2 finite element tetrahedron are related is introduced. A validity condition is then exhibited. Extension to arbitrary degree and dimension is given. A construction method is then proposed and demonstrated by means of various concrete application examples. Copyright © 2012 John Wiley & Sons, Ltd.

Valuation of Loan Credit Default Swaps Correlated Prepayment and Default Risks with Stochastic Recovery Rate

In this paper, we establish an intensity based multi-factor model to value LCDS. The pricing model incorporates the modeling of default, prepayment and recovery risks. Using one factor model, negative correlation between the default and prepayment intensities and positive correlation between the default intensity and the loss given default are described. The interest rate and the house price are chosen as the relevant factors. Under these assumption, a Cauthy problem of PDE is derived, which has a closed-form solution. Based on the solution, numerical examples are provided.

Utility-Based Pricing, Timing and Hedging of an American Call Option Under an Incomplete Market with Partial Information

This paper studies the pricing, timing and hedging of an American call option written on a non-tradable asset whose mean appreciation rate is not observable but is known to be a Gaussian random variable. Our goal is to analyze the effects of the partial information on investment in the American option under an incomplete market. The objective of the option holder is to maximize the expected discounted utility of consumption over an infinite lifetime. Thanks to consumption utility-based indifference pricing principal, stochastic control and filtering theory, under CARA utility, we derive the value and the exercise time of the American call option, which are determined by a semi-closed-form solution of a free-boundary PDE problem with a finite time horizon. We provide numerical results by finite difference methods and compare the results with those under a fully observable case. Numerical calculations demonstrate that partial information leads to a significant loss of the implied value of the American call option. This loss increases with the uncertainty of the mean appreciation rate. If the option holder is risk-averse enough, a growth of the systematic/idiosyncratic risk will increase/decrease the implied option value and the option is exercised late/early. Whether a stronger positive correlation between the tradable asset and the non-tradable underlying asset increases the option value and the information value depends on the risk attitude of the option holder. But on the contrary, a stronger negative correlation will definitely make the option and the information more valuable. In addition, as a corollary of our results, explicit expressions of the utility-based pricing for a perpetual American call are presented if the tradable risky asset is perfectly correlated with the underlying asset.

Finite element simulations of window Josephson junctions

This paper deals with the numerical simulation of the steady state two dimensional window Josephson junctions by finite element method. The model is represented by a sine-Gordon type composite PDE problem. Convergence and error analysis of the finite element approximation for this semilinear problem are presented. An efficient and reliable Newton-preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system. Regular solution branches are computed using a simple continuation scheme. Numerical results associated with interesting physical phenomena are reported. Interface relaxation methods, which by taking advantage of special properties of the composite PDE, can further reduce the overall computational cost are proposed. The implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed.

A collage-based approach to inverse problems for non-linear elliptic PDEs

The collage coding solution strategy for ODE inverse problems, first developed in Kunze and Vrscay (1999), controls the approximation error by bounding it above by a more readily minimisable distance. In this paper, we explore a collage coding technique for solving inverse problems for a general class of second-order non-linear elliptic PDEs. The method is based on the nonlinear Lax-Milgram representation theorem and elements of variational calculus. We develop this method for a broad class of non-linear elliptic PDEs and present an example of the method in practice.

An applied mathematical excursion through Lyapunov inequalities, classical analysis and Differential Equations

Several different problems make the study of the so called Lyapunov type inequalities of great interest, both in pure and applied mathematics. Although the original historical motivation was the study of the stability properties of the Hill equation (which applies to many problems in physics and engineering), other questions that arise in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients, etc. lead to the study of L p Lyapunov inequalities (1 ≤ p ≤ Ω) for differential equations. In this work we review some recent results on these kinds of questions which can be formulated as optimal control problems. In the case of Ordinary Differential Equations, we consider periodic and antiperiodic boundary conditions at higher eigenvalues. Then, we establish Lyapunov inequalities for systems of equations. For Partial Differential Equations on a domain Ω ⊂ ℝN, we consider the Laplace equation with Neumann or Dirichlet boundary conditions. It is proved that the relation between the quantities p and N/2 plays a crucial role in order to obtain nontrivial L p Lyapunov type inequalities (which are called Sobolev inequalities by many authors). One of the main applications of Lyapunov inequalities is its use in the study of nonlinear resonant problems. To this respect, combining the linear results with Schauder fixed point theorem, we show some new results about the existence and uniqueness of solutions for resonant nonlinear problems for ODE or PDE, both in the scalar case and in the case of systems of equations.

Well‐Posed and Ill‐Posed Boundary Value Problems for PDE

Well‐Posed and Ill‐Posed Boundary Value Problems for PDE

PDE problems arising in mathematical biology

This article reviews biological processes that can be modeled by PDEs, it describes mathematical results, and suggests open problems. The first topic deals with tumor growth. This is modeled as a free boundary problem for a coupled system of elliptic, hyperbolic and parabolic equations. Existence theorems, stability of radially symmetric stationary solutions, and symmetry-breaking bifurcation results are stated. Next, a free boundary problem for wound healing is described, again involving a coupled system of PDEs. Other topics include movement of molecules in a neuron, modeled as a system of reaction-hyperbolic equations, and competition for resources, modeled as a system of reaction-diffusion equations.

Best Robin Parameters for Optimized Schwarz Methods at Cross Points

Optimized Schwarz methods are domain decomposition methods in which a large-scale PDE problem is solved by subdividing it into smaller subdomain problems, solving the subproblems in parallel, and iterating until one obtains a global solution that is consistent across subdomain boundaries. Fast convergence can be obtained if Robin conditions are used along subdomain boundaries, provided that the Robin parameters $p$ are chosen correctly. In the case of second-order elliptic problems such as the Poisson equation, it is well known for two-subdomain problems without overlap that the optimal choice is $p = O(h^{-1/2})$ (where $h$ is the mesh size), with the resulting method having a convergence factor of $\rho = 1 - O(h^{1/2})$. However, when cross points are present, i.e., when several subdomains meet at a single point, this choice leads to a divergent method. In this article, we show for a model problem that convergence can occur only if $p = O(h^{-1})$ at the cross point; thus, a different scaling of the Ro...

Numerical strategies for the Galerkin–proper generalized decomposition method

The Proper Generalized Decomposition or, for short, PGD is a tensor decomposition based technique to solve PDE problems. It reduces calculation and storage cost drastically and presents some similarities with the Proper Orthogonal Decomposition, for short POD. In this work, we propose an efficient implementation to improve the convergence of the PGD, toward the numerical solution of a discretized PDE problem, when the associated matrix is Laplacian-like.

Existence and regularity results for fully nonlinear equations with singularities

We consider the Dirichlet boundary value problem for a singular elliptic PDE like F[u] = p(x)u −μ , where p, μ ≥ 0, in a bounded smooth domain of $${\mathbb{R}^n}$$ . The nondivergence form operator F is assumed to be of Hamilton–Jacobi–Bellman or Isaacs type. We establish existence and regularity results for such equations.

Uniqueness of an inverse problem with single measurement data generated by a plane wave in partial finite differences

Uniqueness of the inverse problem for a hyperbolic PDE with the unknown potential and a single incident plane wave is a long-standing, well-known open question. This question is addressed for the case when derivatives with respect to variables, orthogonal to the direction of propagation of this plane wave, are expressed via finite differences.

Min–max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations

We consider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω ⊂ R n , n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min–max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties.

Equivalence problem of second order PDE for scale transformations

Equivalence problem of second order PDE for scale transformations

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

Constitutive law gap functionals for solving the Cauchy problem for linear elliptic PDE

Different numerical methods have been proposed in the literature for solving the Cauchy problem for linear elliptic equations modelling different physical phenomena (Laplace equation for the stationary heat equation, Lamé operator in elasticity, etc.). The aim of this paper is to situate these methods, fixed point methods and domain decomposition based techniques in a general variational framework, and show the equivalence between them. A generalization of the energy gap method proposed by the authors for Fourier-like boundary conditions is studied. Then a comparison of these methods by using an analytical example and two numerical problems with complicated geometry or boundary conditions is performed in order to estimate their numerical performances. It appeared that no method stood out from the others due to better or worse performance. According to the situations, the classification of the performances in terms of conditioning, an essential factor because the Cauchy problem is ill-posed, or in terms of the capacity to deal with strongly spatially variable data, depends on the problem dealt with. The issue of regularization is not addressed here because it is method-dependent and distorts the appreciation of the basic performance of the different approaches.

Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.

A multigrid method for constrained optimal control problems

We consider the fast and efficient numerical solution of linear–quadratic optimal control problems with additional constraints on the control. Discretization of the first-order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints. The complementarity conditions are treated by a primal–dual active set strategy that serves as outer iteration. At each iteration step, a KKT system has to be solved. Here, we develop a multigrid method for its fast solution. To this end, we use a smoother which is based on an inexact constraint preconditioner. We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem. Furthermore, when combined with a nested iteration, the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the PDE solution.

Recent advances in analytical and numerical methods in inverse problems for PDEs (minisymposium report)

Abstract This minisymposium, held at the 5th International Conference “Inverse Problems: Modeling and Simulation”, May 24–29, 2010 in Antalya, Turkey, was dedicated to the 60th birthday of Michael V. Klibanov.

Nonlinear scheme with high accuracy for nonlinear coupled parabolic–hyperbolic system

A nonlinear finite difference scheme with high accuracy is studied for a class of two-dimensional nonlinear coupled parabolic–hyperbolic system. Rigorous theoretical analysis is made for the stability and convergence properties of the scheme, which shows it is unconditionally stable and convergent with second order rate for both spatial and temporal variables. In the argument of theoretical results, difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome, by an ingenious use of the method of energy estimation and inductive hypothesis reasoning. The reasoning method here differs from those used for linear implicit schemes, and can be widely applied to the studies of stability and convergence for a variety of nonlinear schemes for nonlinear PDE problems. Numerical tests verify the results of the theoretical analysis. Particularly it is shown that the scheme is more accurate and faster than a previous two-level nonlinear scheme with first order temporal accuracy.