Incorporating physics into the solving process of partial differential equations can lead to more efficient algorithms. This thesis presents our exploration of physical modeling and development of algorithms in elastic and electromagnetism equations.

In the first part of this thesis, we study the 2+1 dimensional continuum model for the evolution of stepped epitaxial surface under long-range elastic interaction. We propose a modified 2+1 dimensional continuum model based on the underlying physics. This modification fixes the problem of possible illposedness due to the nonconvexity of the energy functional. We prove the existence and uniqueness of both the static and dynamic solutions and derive a minimum energy scaling law for them. We show that the minimum energy surface profile is main...[ Read more ]

Incorporating physics into the solving process of partial differential equations can lead to more efficient algorithms. This thesis presents our exploration of physical modeling and development of algorithms in elastic and electromagnetism equations.

In the first part of this thesis, we study the 2+1 dimensional continuum model for the evolution of stepped epitaxial surface under long-range elastic interaction. We propose a modified 2+1 dimensional continuum model based on the underlying physics. This modification fixes the problem of possible illposedness due to the nonconvexity of the energy functional. We prove the existence and uniqueness of both the static and dynamic solutions and derive a minimum energy scaling law for them. We show that the minimum energy surface profile is mainly attained by surfaces with step meandering instability. We also discuss the transition from the step bunching instability to the step meandering instability in 2+1 dimensions.

In the second part, we propose a novel neural network structure called energy decay convolutional network (EDCnet) for solving general gradient flow equations. We organize the connection of variables for the partial differential equation based on an energy decay scheme while the corresponding operators are learned by convolutional block. Our method is fast, stable, and can be well generalized to other problems. We applied our method to a wide variety of gradient flow equations to demonstrate its promising applications.

Finally, we propose a singular value decomposition based multigrid method for the Helmholtz equation. This method incorporates two key components for efficiency: Krylov subspace as smoother and partial singular value decomposition as assistant of coarse grid correction. The former decreases the amplification brings from the classical smoothers while the later alleviates the error caused by mismatched eigenvalues between fine and coarse levels. The numerical results illustrate the power of this iterative-direct mixed method, especially for scenarios when the spectral distribution is relatively poor.