In this paper, a stochastic finite element approximation scheme is developed for an optimal control problem governed by an elliptic integro-differential equation with random coefficients. Different from the well-studied optimal control problems governed by stochastic PDEs, our control problem has the control constraints of obstacle type, which is mostly seen in real applications. We develop the weak formulation for this control and its stochastic finite element approximation scheme. We then obtain necessary and sufficient optimality conditions for the optimal control and the state, which are the base for deriving a priori error estimates of the approximation in our work. Instead of using the infinite dimensional Lagrange multiplier theory, which is currently used in the literature but often difficult to handle inequality control constraints, we use a direct approach by applying the well-known Lions' Lemma to the reduced optimal problem. This approach is shown to be applicable for a wide range of control constraints. Finally numerical examples are presented to illustrate our theoretical results.

In this article, a characteristic scheme is considered for the viscoelastic Oldroyd fluid flows based on the lowest equal-order finite element pair. The diffusion term in these equations is discretized by using finite element method, the temporal differentiation and advection terms are treated by characteristic scheme and the integral term is handled by applying right rectangle rule. Unconditionally stability and optimal error estimates for the velocity and pressure are derived. Finally, some numerical results are provided to verify the performance of the proposed method.

This paper is devoted to the study of a fully discrete $T-\psi$ 8finite element method based on $H$-decomposition to solve a nonlinear degenerate transient eddy current problem with ferromagnetic materials. Here, the ferromagnetic properties are linked by a power material law. We first design a nonlinear time-discrete scheme for approximation in suitable function spaces. We show the well-posedness of the semidiscrete problem and prove the convergence of the nonlinear scheme by the Minty-Browder technique. Finally, we suggest a fully discrete scheme, derive its error estimate and give some numerical experiments to validate the theoretical result.

In this paper, we investigate a spectral method for mixed boundary value problems defined on hexahedrons. Some results on irrational orthogonal approximation are established, which play important roles in numerical solutions of partial differential equations defined on hexahedrons. As examples of applications, we provide spectral schemes for two model problems, and prove their spectral accuracy. Efficient numerical implementations are described. Numerical results demonstrate the high efficiency of suggested algorithms.

The re-weighted regularized conjugate gradient (RRCG) method has been a popular algorithm for magnetic inversion problems. In this work, we show that for a two-dimensional problem with uniform field data, the resulting coefficient matrix to be inverted has a symmetric Block-Toeplitz Toeplitz-Block (BTTB) structure. Taking advantage of the BTTB properties, the storage and computational complexity can be significantly reduced, so that the efficiency of the RRCG method is greatly improved and it is now capable of dealing with much larger system with a modest computing resource. This paper also investigates various numerical inversion schemes including the CG type and multigrid (MG) methods. It has been demonstrated that the MG is an efficient and robust numerical tool for magnetic field inversion. Not only the MG produces a rapid convergence rate, the performance is not sensitive when applying to noisy data. Numerical simulations using synthetic data and real field data are reported to confirm the effectiveness of the MG method.

We propose and analyze a partitioned numerical method for the fully evolutionary Stokes-Darcy equations that model the coupling between surface and groundwater flows. The proposed method uncouples the surface from the groundwater flow by using the implicit-explicit combination of the Crank-Nicolson and Leapfrog methods for the discretization in time with added stabilization terms. We prove that the method is asymptotically, unconditionally stable — requiring no time step condition — and second-order accurate in time with optimal rates in space. We verify the method's unconditional stability and second-order accuracy numerically.

The existence and uniqueness of a vector scalar potential representation with the Lorenz gauge (Schelkunoff potential) is proven for any vector field from H(curl). This representation holds for electric and magnetic fields in the case of a piecewise smooth conductivity, permittivity and permeability, for any frequency. A regularized formulation for the magnetic field is obtained for the case when the magnetic permeability $\mu$ is constant and thus the magnetic field is divergence free. In the case of a non divergence free electric field, an equation involving scalar and vector potentials is proposed. The solution to both electric and magnetic formulations may be approximated by the nodal shape functions in the finite element method with system matrices that remain well-conditioned for low frequencies. A numerical study of a forward problem of a computation of electromagnetic fields in the diffusive electromagnetic regime shows the efficiency of the proposed method.

A family of interior penalty $hp$-discontinuous Galerkin methods is developed and analyzed for the numerical solution of the quasilinear elliptic equation $-\nabla \cdot (\rm{A}$ $(\nabla u)\nabla u)=f$ posed on the open bounded domain $\Omega\subset\mathbb{R}^d$, $d\geq2$. Subject to the assumption that the map $\rm{v}\mapsto \rm{A}(\rm{v})\rm{v}$, $\rm{v} \in \mathbb{R}^d$, is Lipschitz continuous and strongly monotone, it is proved that the proposed method is well-posed. A priori error estimates are presented of the error in the broken $H^1(\Omega)$-norm, exhibiting precisely the same $h$-optimal and mildly $p$-suboptimal convergence rates as obtained for the interior penalty approximation of linear elliptic problems. A priori estimates for linear functionals of the error and the $L^2(\Omega)$-norm of the error are also established and shown to be $h$-optimal for a particular member of the proposed family of methods. The analysis is completed under fairly weak conditions on the approximation space, allowing for non-affine and curved elements with multilevel hanging nodes. The theoretical results are verified by numerical experiments.