Laboratory of Applied Mathematics, SHIGETA

Action mechanism of yieldin

1. Highly accurate and fast numerical computations for direct problems of partial differentialequations
2. Numerical computations for inverse problems of partial differential equations

Direct problems

The method of fundamental solutions (MFS) has recently been popular with people who wouldlike to easily solve partial differential equations. The MFS is a truly mesh-free method of boundarytype. Parameters used for the MFS should suitably be chosen to obtain accurate solutions for theproblem to be considered. In addition, it is also important to reduce the condition number of amatrix derived from the MFS to obtain stable solutions since the resultant matrix equation is ill-conditioned.

We consider the MFS for solving the Dirichlet problem of the Laplace equation or the Stokesequation in an exterior unbounded domain. One of advantages of the MFS is to directly solveexterior problems, which cannot directly be solved by the finite element method (FEM) and othermesh-free methods using radial basis functions. Even though the basis functions used in the MFSdo not satisfy a condition that the solution converges to zero at infinity, the accuracy is fairly goodin the whole computational domain. In the study, the approximate solution for the exteriorproblems is defined by a linear combination of the proper basis functions satisfying the governingequation as well as the condition at infinity according to the approach of Katsurada.

In the case of the Laplace equation, the numerical solution obtained by the MFS is accurate,while the corresponding matrix equation is ill-conditioned. Hence, a modified MFS (MMFS) withthe proper basis functions is proposed by the introduction of the modified Trefftz method (MTM).The concrete expressions of the corresponding condition numbers are given in mathematical formsand the solvability by these methods is mathematically proven. Thereby, the optimal parameterminimizing the condition number is also mathematically given. Numerical experiments show thatthe condition numbers of the matrices corresponding to the MTM and the MMFS are drasticallyreduced and that the numerical solution by the MMFS is more accurate than the one by theconventional method.

The case of the Stokes equation is now under study.

Many kinds of inverse problems have been studied in science and engineering. The Laplaceequation is a famous governing equation of many physical phenomena as well as a mathematicallybasic equation. We consider the Cauchy problem of the Laplace equation. This is a well knowninverse boundary value problem, where for given boundary data as a priori information on a partof the boundary of a domain, we identify unknown boundary value on the rest of the boundary.This problem can be regarded as mathematical models of inverse problem of electrocardiographyin medical science as well as the defect identification for the corrosion of a blast furnace bythermal observations.

We consider a problem with noisy Cauchy data, since the Cauchy data usually contain noisesdue to observation errors. In past years, some researchers have numerically solved the Cauchyproblem by various methods. However, to our knowledge, their methods, which are the finitedifference method or the spectral collocation method in multiple-precision arithmetic, cannotsuccessfully solve a problem whose exact solution has singular points outside the computationaldomain. We use the method of fundamental solutions (MFS) to directly discretize the Cauchyproblem of the Laplace equation. The Cauchy problem is an ill-posed problem, that is, the solutiondoes not depend continuously on the given boundary data. Namely, a small noise contained in thegiven Cauchy data has a possibility to affect sensitively on the accuracy of the solution.Consequently, we use the Tikhonov regularization to avoid the sensitivity and to obtain a stableregularized solution. The regularized solution depends on a regularization parameter. Then, we usethe L-curve suggested by Hansen for determining a suitable regularization parameter to obtain abetter regularized solution. We numerically indicate that a suitable regularized solution obtainedby the L-curve is optimal in the sense that the error between the regularized solution and the exactone is minimized. We respectively show the accuracy and the optimal regularization parameteragainst a noise level. We also mention influence of the total numbers of the source and thecollocation points, which are used in the MFS, on accuracy. Based on our numerical experiments,it is concluded that the numerical method proposed in the study is effective for a problem whosesolution has singular points outside the computational domain. No multiple-precision arithmetic isrequired to obtain a good solution. It is noteworthy that such kind of problems can alsosuccessfully be solved. Moreover, the method is applicable for solving a problem in a complicateddomain with the Cauchy data that contains large noises even with a noise level of 10%.