In this paper, we analyze implicit-explicit (IMEX) Runge-Kutta (RK) time discretization methods for solving linear convection-diffusion equations. The diffusion operator is treated implicitly via the embedded discontinuous Galerkin (EDG) method and the convection operator explicitly via the upwinding discontinuous Galerkin method.

In this article, we consider the Poisson equation on a polygonal domain with the domain singularity raised from the changed boundary conditions with the inner angle $\omega>\frac{\pi}{2}$. The solution of the Poisson equation with such singularity has a singular decomposition: regular part plus singular part. The singular part is a linear combination of one or two singular functions. The coefficients of the singular functions are usually called stress intensity factors and can be computed by the extraction formula. In [11] we introduced a new partial differential equation which has 'zero' stress intensity factor using this stress intensity factor, from whose solution we can obtain a very accurate solution of the original problem simply by adding singular part. Although the method in [11] works well for the Poisson problem with Dirichlet boundary condition, it does not give optimal results for the case with stronger singularity, for example, mixed boundary condition with bigger inner angle. In this paper we give a revised algorithm which gives optimal convergences for both cases.

In this paper, we study a third order accurate linearized backward differential formula (BDF) type scheme for the nonlinear Maxwell's equations, using the Nédelec finite element approximation in space. A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. An optimal $L^2$ error estimate is presented, via a linearized stability analysis for the numerical error function, under a condition for the time step, $\tau \leq C^*_0h^2$ for a fixed constant $C^*_0$. Numerical results are provided to confirm our theoretical analysis and demonstrate the high order accuracy and stability (convergence) of the linearized BDF finite element method.

We propose a new finite element method for solving second order elliptic interface problems whose solution has a Robin type jump along the interface. We cast the problem into a new variational form and introduce a finite element method to solve it using a uniform grid. We modify the $P_1$-Crouzeix-Raviart element so that the shape functions satisfy the jump conditions along the interface. We note that there are cases that the Lagrange type basis can not be used because of the jump in the value. Numerical experiments are provided.

In this paper, we propose several overlapping domain decomposition preconditioners for solving the unconstrained elliptic optimal control problem, based on the two level additive Schwarz algorithm. We consider the cases with controls on the whole domain and controls from a local subset. The latter case can be viewed as the subproblems when we solve the control-constrained control problem by using semi-smooth Newton method. When the controls act on the whole domain, we construct a symmetric and positive definite preconditioner which is proved to be robust combined with preconditioned MINRES method, and a symmetric and indefinite preconditioner which can be used in the preconditioned GMRES method and shows better numerical performance than the positive definite one. When the controls act on a local subset, we also construct a similar symmetric and indefinite preconditioner, the numerical experiments show its efficiency when combined with preconditioned GMRES method.

In this article, a posteriori error analysis for mixed finite element Galerkin approximations of second order linear hyperbolic equations is discussed. Based on mixed elliptic reconstructions and an integration tool, which is a variation of Baker's technique introduced earlier by G. Baker (SIAM J. Numer. Anal., 13 (1976), 564-576) in the context of a priori estimates for a second order wave equation, a posteriori error estimates of the displacement in $L^∞(L^2)$-norm for the semidiscrete scheme are derived. Finally, a first order implicit-in-time discrete scheme is analyzed and a posteriori error estimators are established.

In this paper, we introduce a simple method for the Cauchy problem. This new finite element method is based on least squares methodology with discontinuous approximations which can be implemented and analyzed easily. This discontinuous Galerkin finite element method is flexible to work with general unstructured meshes. Error estimates of the finite element solution are derived. The numerical examples are presented to demonstrate the robustness and flexibility of the proposed method.

We present a least squares framework for constructing $p$-th degree immersed finite element (IFE) spaces for typical second-order elliptic interface problems. This least squares formulation enforces interface jump conditions including extended ones already proposed in the literature, and it guarantees the existence of $p$-th IFE shape functions on interface elements. The uniqueness of the proposed $p$-th degree IFE shape functions is also discussed. Computational results are presented to demonstrate the approximation capabilities of the proposed $p$-th IFE spaces as well as other features.

In this paper, we investigate a stochastic spline-collocation approximation scheme for an optimal control problem governed by an elliptic PDE with random field coefficients. We obtain the necessary and sufficient optimality conditions for the optimal control problem and establish a scheme to approximate the optimality system through the discretization with respect to the spatial space by finite elements method and the probability space by stochastic spline-collocation method. We further investigate Smolyak approximation schemes, which are effective collocation strategies for smooth problems that depend on a moderately large number of random variables. For more general control problems where the state may be non-smooth with respect to the random variables in some areas, we adopt a domain decomposition strategy to partition the random space into smooth and non-smooth parts and then apply Smolyak scheme and spline approximation respectively. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.

We develop a finite element method for solving the Dirichlet problem of the one- dimensional prescribed curvature equation due to its irreplaceable role in applications. Specifically, we first analyze the existence and uniqueness of the solution of the problem and then develop a finite element method to solve it. The well-posedness of the finite element method is shown by employing the Banach fixed-point theorem. The optimal error estimates of the proposed method in both the $H^1$ norm and the $L^2$ norm are established. We also design a Newton type iteration scheme to solve the resulting discrete nonlinear system. Numerical experiments are presented to confirm the order of convergence of the proposed method.

The paper is concerned with $L^p$ error analysis of semi-discrete Galerkin FEMs for nonlinear parabolic equations. The classical energy approach relies heavily on the strong regularity assumption of the diffusion coefficient, which may not be satisfied in many physical applications. Here we focus our attention on a general nonlinear parabolic equation (or system) in a convex polygon or polyhedron with a nonlinear and Lipschitz continuous diffusion coefficient. We first establish the discrete maximal $L^p$-regularity for a linear parabolic equation with time-dependent diffusion coefficients in $L^∞(0,T;W^{1,N+\epsilon}) \cap C(\overline{\Omega} \times [0,T])$ for some $\epsilon>0$, where $N$ denotes the dimension of the domain, while previous analyses were restricted to the problem with certain stronger regularity assumption. With the proved discrete maximal $L^p$-regularity, we then establish an optimal $L^p$ error estimate and an almost optimal $L^∞$ error estimate of the finite element solution for the nonlinear parabolic equation.

In this paper, a nonuniform size modified Poisson-Boltzmann equation (SMPBE) for a protein in a solvent with multiple ionic species in distinct ionic sizes is derived by using a new electrostatic free energy functional and solution decomposition techniques. It is then proved to have a unique solution, and the solution satisfies a system consisting of nonlinear algebraic equations and one Poisson dielectric interface problem. To solve it numerically, two new finite element iterative schemes are proposed by using nonlinear successive over-relaxation techniques, along with an improved uniform SMPBE for generating initial iterates. Furthermore, they are programmed in Python and Fortran as a software package for solving the nonuniform SMPBE, and numerically tested on a Born ball test model and a protein in a sodium chloride solution and a sodium chloride and potassium chloride solution. Numerical results confirm the convergence of the two new iterative schemes and demonstrate the high performance of the new software package.

A second-order accurate augmented method is proposed and analyzed in this paper for a general elliptic PDE with a general boundary condition using the jump embedded boundary conditions (JEBC) formulation. First of all, the existence and uniqueness of an interface problem with given are discussed. Then, the well-posedness theory is extended to the interface problems with given jump conditions. In the proposed numerical method, one novel idea is to preconditioning the PDE first so that the coefficient of the highest derivative is of $O(1)$. The second idea is to introduce two augmented variables corresponding to the jump in the solution and its normal derivative along the boundary to get an interface problem. For a piecewise constant coefficient, the fast Poisson solver then can be utilized in a rectangular domain. The augmented variables can be determined from a Schur complement system. We also propose two preconditioning techniques for the GMRES iterative method for the Schur complement; one is from the flux jump condition, and the other one is from the algebraic preconditioner based on the interpolation scheme in the augmented algorithm. The presented numerical results show that the proposed method has not only obtained second order accurate solutions in the $L^∞$ norm globally, but also second order accurate normal derivatives at the boundary from each side of the interface. The proposed preconditioning technique can speed up 50-90% compared with the method without preconditioning.

We propose a 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations. The low-order conforming finite elements are not inf-sup stable. The low-order nonconforming finite elements do not satisfy the Korn inequality. The proposed finite element space consists of two conforming components and one nonconforming component. We prove that the discrete inf-sup condition is valid and the discrete Korn inequality holds uniformly in the mesh-size. Based on these results we give some numerical verification. In addition, this element is compared numerically with six other mixed finite elements.

In this paper, an $hp$ hybridizable weak Galerkin ($hp$-HWG) method is introduced to solve the Helmholtz equation with large wave number in two and three dimensions. By choosing a specific parameter and using the duality argument, we prove that the proposed method is stable under certain mesh constraint. Error estimate is obtained by using the stability analysis and the duality argument. Several numerical results are provided to confirm our theoretical results.

We develop a sparse grid spectral element method using nodal bases on Chebyshev-Gauss-Lobatto points for multi-dimensional elliptic equations. Since the quadratures based on sparse grid points do not have the accuracy of a usual Gauss quadrature, we construct the mass and stiffness matrices using a pseudo-spectral approach, which is exact for problems with constant coefficients and uniformly structured grids. Compared with the regular spectral element method, the proposed method has the flexibility of using a much less degree of freedom. In particular, we can use less points on edges to form a much smaller Schur-complement system with better conditioning. Preliminary error estimates and some numerical results are also presented.

In this paper, we propose and analyze the local discontinuous Galerkin method for the viscous Burgers-Poisson system. The proposed method preserves two invariants and hence, yields solutions even for long time. A priori error estimates, which are of order $\mathcal{O}(h^{k+1})$, when polynomials of degree $k\geq 1$ are used for approximating solutions are established. Finally, numerical experiments are conducted to confirm our theoretical results.