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Mitsuhiro Nakao

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Numerical Verification Method of Solutions for Partial Differential Equations

Mitsuhiro Nakao
Degree: Doctor of Science (Kyushu University)
Research Interests: Computational Mathematics
Unit:Number & Equation

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My current research topic is “numerical verification method for solutions to partial differential equations”. Partial differential equations are widely used as mathematical models in numerical simulations in science, engineering, and industry, and determining their solutions is important in industrial technologies. However, the gap of infinite-dimensional space must be overcome to find mathematically rigorous solutions, and the theoretical and numerical difficulties are forcing much of the real world to accept approximate solutions. 。

My group strives to achieve computer assisted proofs for the existence of rigorous solutions to partial differential equations. In 1988, I was the first to effectively combine the finite element approximation in the elliptic boundary value problem and its constructive a priori error estimation with interval analysis. Our research demonstrated that a rigorous solution can be captured using a computer. Today this method is known as “Nakao’s method”. By collaborating with other researchers, my group has successfully conducted research based on the same principle to achieve a verification precision, efficiency and range of applicable region of the Dirichlet problem for second-order semilinear equations to a practically useful level. This achievement on the Dirichlet problem is attracting international attention from a quality assurance point of view of computation as a rigorous a posteriori error estimation for an approximate calculation as well as from the viewpoint of a computer-assisted proof for an analytical problem, which is theoretically difficult to solve.

We have been broadening our research to include steady-state Navier-Stokes equations and variational inequality as well as eigenvalue problems and inverse eigenvalue problems of elliptic operators. Recently, we worked on a computer-assisted proof for the existence of a bifurcating solution to non-linear heat convection under the Navier-Stokes equation (Rayleigh-Benard problem), which is theoretically difficult to analyze. Specifically, we pursued a bifurcating curve with verification in twodimensional space of Rayleigh-Benard problem as well as formulated a verification of the bifurcation point itself. We were able to demonstrate with mathematical rigor that symmetry-breaking bifurcation actually occurs on a bifurcation curve using a self-validating numerical method. Furthermore, we successfully employed numerical verification to determine the formation of hexagonal patterns, which is extremely interesting as a three-dimensional heat convection phenomenon (Please see the figure). Because each of these is a difficult problem to solve using a conventional theoretical approach, these are important results for computer-assisted proofs.

Additionally, we have successfully derived a method to numerically determine various constants that appear in a constructive a priori error estimation of the finite element method using a computer calculation to guarantee precise values, which are difficult to theoretically determine.

In numerical simulation used in industry, various numerical computations (scientific computations) are used, including approximate solutions to differential equations. From a mathematical point of view, there are numerous methods with questionable calculation precision, and this precision is the reason for discrepancy between simulation results and actual phenomena. Hence, it is important to reevaluate these simulation methods from a mathematical reliability point of view in order to guarantee the precision of calculations and to improve the precision of numerical simulation.

Today numerous issues have yet to be solved to completely control the precision of solutions to a broad range of partial differential equations, which are significant in engineering and industry. Hence, my group continues with our research. However, I am optimistic that a rapid increase in computer performance as well as theoretical and practical improvements of verification methods will one day lead to precision guarantee.

Moreover, I would like to challenge the limit of rigorous mathematical discussion using a computer for problems with infinite dimensions from a mathematical point of view. I strive to launch “computer assisted analysis” as constructive mathematics from a numerical computation point of view in the future.

Fig.1:Heatconvection phenomenon called a honeycomb pattern (experiment)

Fig.2:‘‘Numerical solutions with guaranteed precision” to the heat convection problem

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