• Abstract

We establish an unconditional and optimal strong convergence rate of Wong–Zakai type approximations in Banach space norm for a parabolic stochastic partial differential equation with monotone drift, including the stochastic Allen–Cahn equation, driven by an additive Brownian sheet. The key ingredient in the analysis is the full use of additive nature of the noise and monotonicity of the drift to derive a priori estimation for the solution of this equation. Then we use the factorization method and stochastic calculus in martingale type 2 Banach spaces to deduce sharp error estimation between the exact and approximate Ornstein–Uhlenbeck processes, in Banach space norm. Finally, we combine this error estimation with the aforementioned a priori estimation to deduce the desired strong convergence rate of Wong–Zakai type approximations.

• Abstract

The aim of this paper is to propose and analyze a numerical method to solve a timedependent eddy current problem in a domain containing moving non magnetic conductors. To this end, we choose a formulation in terms of the magnetic field, what leads to a parabolic problem for which we prove an existence result. For space discretization, we propose a finite element method based on Nédélec edge elements on a mesh that remains fixed over the time. The curl-free constraint in the dielectric domain is relaxed by means of a penalty strategy that can be easily implemented, without the need that the mesh fits the moving conducting and dielectric domains. For time discretization, we use a backward Euler scheme. We report some numerical results. First, we solve a test problem with a known analytical solution, which allows us to assess the convergence of the method as the penalization and discretization parameters go to zero. Finally, we solve a problem with cylindrical symmetry, which allows us to compare the results with those obtained with an axisymmetric code.

• Abstract

In this paper we propose several postprocessing techniques to accelerate the convergence of the continuous Galerkin solutions for delay differential equations with nonlinear vanishing delay. They are interpolation postprocessings (including integration type, Lagrange type, and polynomial preserving recovery type) and iteration postprocessing. The theoretical expectations are confirmed by numerical experiments.

• Abstract

We consider an optimal boundary control problem described by a semilinear parabolic partial differential equation, with control and state constraints. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized by using a finite element method in space and a partially implicit scheme in time, while the controls are approximated by piecewise constant relaxed controls. We first state the necessary conditions for optimality for the continuous problem and the discrete relaxed problem. Next, under appropriate assumptions, we prove that accumulation points of sequences of optimal (resp. admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal) for the continuous relaxed problem.

• Abstract

A new anisotropic adaptive mesh refinement method based on anisotropic bubbletype local mesh generation (ABLMG) for elliptic partial differential equations is proposed. The anisotropic meshes are generated as quasi-uniform meshes in metric spaces with the metric determined on each vertex by anisotropic a posteriori error estimator. Under the new metric, the error is equidistributed in the directions of maximum and minimum stretching on an element, and the mesh size is reduced/coarsend in regions with large/small errors. With the full use of the adjacent lists provided by the node placement method, the local mesh for each vertex is generated through ABLMG method. Compared with other methods, the mesh refining and coarsening can be obtained in the same framework and the mesh suits the metric well at each refinement level. Numerical results in two-dimensions are presented to verify the ability of our metric tensor to generate anisotropic mesh with correct concentration and stretching direction.

• Abstract

Because of the discontinuity of the interface problems, it is natural to apply the discontinuous Galerkin (DG) finite element methods to solve those problems. In this work, both fitted and unfitted mixed hybrid discontinuous Galerkin (MHDG) finite element methods are proposed to solve the elliptic interface problems. For the fitted case, the problems can be solved directly by MHDG method. For the unfitted case, the broken basis functions (unnecessary to satisfy the jump conditions) are introduced to those elements which are cut across by interface, the weights depending on the volume fractions of cut elements and the different diffusions (or material heterogeneities) are used to stabilize the method, and the idea of the Nitsche's penalty method is applied to guarantee the jumps on the interface parts of cut elements. Unlike the immersed interface finite element methods (IIFEM), the two jump conditions are enforced weakly in our variational formulations. So, our unfitted interface MHDG method can be applied more easily than IIFEM to general cases, particularly when the immersed basis function cannot be constructed. Numerical results on convergence and sensitivities of both interface location within a cut element and material heterogeneities show that the proposed methods are robust and efficient for interface problems.

• Abstract

An efficient multigrid method is proposed to compute the ground state solution of Bose-Einstein condensations by the finite element method based on the combination of the multigrid method for nonlinear eigenvalue problem and an efficient implementation for the nonlinear iteration. The proposed numerical method not only has the optimal convergence rate, but also has the asymptotically optimal computational efficiency which is independent from the nonlinearity of the problem. The independence from the nonlinearity means that the asymptotic estimate of the computational work can reach almost the same as that of solving the corresponding linear boundary value problem by the multigrid method. Some numerical experiments are provided to validate the efficiency of the proposed method.

• Abstract

We consider an augmented mixed finite element method for the equations of plane linear elasticity with mixed boundary conditions. The method provides simultaneous approximations of the displacements, the stress tensor and the rotation. We develop an a posteriori error analysis based on the Ritz projection of the error and the use of an appropriate auxiliary function, and derive fully local reliable a posteriori error estimates that are locally efficient up to the elements that touch the Neumann boundary. We provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice.

• Abstract

We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the stability and $hp$-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new inverse inequality for a special class of polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with an $arbitrary$ number of faces for polynomial basis with degree $p$ = 2, 3. The key feature of the proposed method is that it employs elemental polynomial bases of total degree $\mathcal{P}$_$p$, defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. A series of numerical experiments are presented to demonstrate the performance of the proposed DGFEM on general polygonal/polyhedral meshes.