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A fast adaptive numerical solver for nonseparable elliptic partial differential equations
Lee, June-Yub 한국산업정보응용수학회 1998 한국산업정보응용수학회 Vol.2 No.1
We describe a fast numerical method for non-separable elliptic equations in self-adjoin form on irregular adaptive domains. One of the most successful results in numerical PDE is developing rapid elliptic solvers for separable EPDEs, for example, Fourier transformation methods for Poisson problem on a square, however, it is known that there is no rapid elliptic solvers capable of solving a general nonseparable problems [8]. It is the purpose of this paper to present an iterative solver for linear EPDEs in self-adjoint form. The scheme discussed in this paper solves a given non-separable equation using a sequence of solutions of Poisson equations, therefore, the most important key for such a method is having a good Poison solver. High performance is achieved by using a fast high-order adaptive Poisson solver which requires only about 500 floating point operations per gridpoint in order to obtain machine precision for both the computed solution and its partial derivatives. A few numerical examples have been presented.
GLOBAL SHAPE OF FREE BOUNDARY SATISFYING BERNOULLI TYPE BOUNDARY CONDITION
Lee, June-Yub,Seo, Jin-Keun Korean Mathematical Society 2000 대한수학회지 Vol.37 No.1
We study a free boundary problem satisfying Bernoulli type boundary condition along which the gradient of a piecewise harmonic solution jumps zero to a given constant value. In such problem, the free boundary splits the domain into two regions, the zero set and the harmonic region. Our main interest is to identify the global shape and the location of the zero set. In this paper, we find the lower and the upper bound of the zero set. In a convex domain, easier estimation of the upper bound and faster disk test technique are given to find a rough shape of the zero set. Also a simple proof on the convexity of zero set is given for a connected zero set in a convex domain.
A NOTE ON OPTIMAL RECONSTRUCTION OF MAGNETIC RESONANCE IMAGES FROM NON-UNIFORM SAMPLES IN k-SPACE
JUNE-YUB LEE 한국산업응용수학회 2010 Journal of the Korean Society for Industrial and A Vol.14 No.1
A goal of Magnetic Resonance Imaging is reproducing a spatial map of the effective spin density from the measured Fourier coefficients of a specimen. The imaging procedure can be done by inverse Fourier transformation or backward fast Fourier transformation if the data are sampled on a regular grid in frequency space; however, it is still a challenging question how to reconstruct an image from a finite set of Fourier data on irregular points in k-space. In this paper, we describe some mathematical and numerical properties of imaging techniques from non-uniform MR data using the pseudo-inverse or the diagonal-inverse weight matrix. This note is written as an easy guide to readers interested in the non-uniform MRI techniques and it basically follows the ideas given in the paper by Greengard-Lee-Inati [10, 11].
A NUMERICAL METHOD FOR CAUCHY PROBLEM USING SINGULAR VALUE DECOMPOSITION
Lee, June-Yub,Yoon, Jeong-Rock Korean Mathematical Society 2001 대한수학회논문집 Vol.16 No.3
We consider the Cauchy problem for Laplacian. Using the single layer representation, we obtain an equivalent system of boundary integral equations. We show the singular values of the ill-posed Cauchy operator decay exponentially, which means that a small error is exponentially amplified in the solution of the Cauchy problem. We show the decaying rate is dependent on the geometry of he domain, which provides the information on the choice of numerically meaningful modes. We suggest a pseudo-inverse regularization method based on singular value decomposition and present various numerical simulations.
A second order operator splitting method for Allen–Cahn type equations with nonlinear source terms
Lee, Hyun Geun,Lee, June-Yub Elsevier 2015 PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIO Vol.432 No.-
<P><B>Abstract</B></P> <P>Allen–Cahn (AC) type equations with nonlinear source terms have been applied to a wide range of problems, for example, the vector-valued AC equation for phase separation and the phase-field equation for dendritic crystal growth. In contrast to the well developed first and second order methods for the AC equation, not many second order methods are suggested for the AC type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically. In this paper, we propose a simple and stable second order operator splitting method. A core idea of the method is to decompose the original equation into three subequations with the free-energy evolution term, the heat evolution term, and a nonlinear source term, respectively. It is important to combine these three subequations in proper order to achieve the second order accuracy and stability. We propose a method with a half-time free-energy evolution solver, a half-time heat evolution solver, a full-time midpoint solver for the nonlinear source term, and a half-time heat evolution solver followed by a final half-time free-energy evolution solver. We numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth.</P> <P><B>Highlights</B></P> <P> <UL> <LI> The proposed operator splitting method is second order accurate and stable. </LI> <LI> The method can be easily applicable to wide class of Allen–Cahn type equations with non-linear source terms. </LI> <LI> Simulations for multi-component phase separation and dendrite growth show the feasibility of the method. </LI> </UL> </P>
First- and second-order energy stable methods for the modified phase field crystal equation
Lee, Hyun Geun,Shin, Jaemin,Lee, June-Yub Elsevier 2017 Computer methods in applied mechanics and engineer Vol.321 No.-
<P><B>Abstract</B></P> <P>The phase field crystal (PFC) model was extended to the modified phase field crystal (MPFC) model, which is a sixth-order nonlinear damped wave equation, to include not only diffusive dynamics but also elastic interactions. In this paper, we present temporally first- and second-order accurate methods for the MPFC equation, which are based on an appropriate splitting of the energy for the PFC equation. And we use the Fourier spectral method for the spatial discretization. The first- and second-order methods are shown analytically to be unconditionally stable with respect to the energy and pseudoenergy of the MPFC equation, respectively. Numerical experiments are presented demonstrating the accuracy and energy stability of the proposed methods.</P> <P><B>Highlights</B></P> <P> <UL> <LI> We present temporally first- and second-order accurate methods for the MPFC equation. </LI> <LI> The unconditional stability of the proposed methods is analytically proven. </LI> <LI> The proposed methods represent a good balance between accuracy and energy stability. </LI> </UL> </P>