Steady-state solution of the PTC thermistor problem using a quadratic spline finite element method.
Stick-slip transition capturing by using an adaptive finite element method
The numerical modeling of the fully developed Poiseuille flow of a newtonian fluid in a square section with slip yield boundary condition at the wall is presented. The stick regions in outer corners and the slip region in the center of the pipe faces are exhibited. Numerical computations cover the complete range of the dimensionless number describing the slip yield effect, from a full slip to a full stick flow regime. The resolution of variational inequalities describing the flow is based on the...
Stick-slip transition capturing by using an adaptive finite element method
The numerical modeling of the fully developed Poiseuille flow of a Newtonian fluid in a square section with slip yield boundary condition at the wall is presented. The stick regions in outer corners and the slip region in the center of the pipe faces are exhibited. Numerical computations cover the complete range of the dimensionless number describing the slip yield effect, from a full slip to a full stick flow regime. The resolution of variational inequalities describing the flow is based on the...
Strengthened Cauchy-Schwarz and Hölder inequalities.
Strong superconvergence of finite element methods for linear parabolic problems.
Study of convergence of the projection-difference method for hyperbolic equations.
Subjective surfaces and Riemannian mean curvature flow of graphs.
Superconvergence estimates of finite element methods for American options
In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation...
Superconvergence of finite element method for parabolic problem.
Superconvergence of mixed finite element semi-discretizations of two time-dependent problems
We will show that some of the superconvergence properties for the mixed finite element method for elliptic problems are preserved in the mixed semi-discretizations for a diffusion equation and for a Maxwell equation in two space dimensions. With the help of mixed elliptic projection we will present estimates global and pointwise in time. The results for the Maxwell equations form an extension of existing results. For both problems, our results imply that post-processing and a posteriori error estimation...
Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems
The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handle easily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh,...
Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems
The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handle easily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh,...
Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell’s equations
In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order...
Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell’s equations∗
In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order...
The discontinuous Galerkin method for semilinear parabolic problems
The existence of a solution and a numerical method for the Timoshenko nonlinear wave system
The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version of...
The existence of a solution and a numerical method for the Timoshenko nonlinear wave system
The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version...
The fictitious domain method for the mixed finite element approximation of the wave equation
The finite element method with nodes moving along the characteristics for convection-diffusion equations.
The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type
We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set , the gradient approximation possesses the superconvergence: , where denotes the average gradient on elements containing vertex . Furthermore, by using the interpolation post-processing technique,...