Parallel Multiscale FEM Algorithm Of Parabolic Equation For Composite Materials With Periodic Configuration

Parabolic equation has a widespread use in many problems, such as sound heat propa-gation, elastic vibration, hydrogeology, oil production. It is also involved in natural sciencelike statistical physics, probability theory, quantum mechanics, biochemistry and so on.Thus, it is particularly important to study the numerical methods of parabolic equation.In this article, we introduce integral transformation in multiscale asymptotic analysisof parabolic equation in composite materials with periodic configuration. A time-domainparallel numerical multiscale FEM method is introduced.The Fourier transform in time is first applied to obtain a set of complex-valued el-liptic problems in frequency domain, then the multiscale asymptotic analysis is presentedand multiscale asymptotic solutions are obtained in frequency domain. The inverse discreteFourier transform will then recover the approximate solution in time domain. We can solvepartial diferential equations at diferent frequency points in frequency domain, so the algo-rithm is parallel in time.The main contents in this article include five parts. Multiscale asymptotic solutions offrequency equation in frequency domain is obtained by multiscale asymptotic analysis. Themultiscale truncation error estimate of frequency equation on bounded smooth domain isgiven. By constructing boundary layer, the multiscale truncation error estimate of frequencyequation on bounded polygonal convex domain is also given. Full error estimates are derivedby combining the inverse integral transform. Finally, numerical examples are given to showthe validity and the accuracy of the presented method.