This paper extends previous results on nonlinear Schwarz preconditioning (Cai and Keyes 2002) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The nonlocal finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in Jones...

An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region$\Omega$ (E) ${\left(t{u}_t\right)}_t={\sum }_{i,j=1}{\left(a_{ij}\left(x\right)u_{x_i}\right)}_{x_j}-a_0\left(x\right)u+f\left(u\right)$, subject to the initial and boundary conditions, $u=0$ on $\partial \Omega$ and $u(x,0)=u_0$. (E) is degenerate at $t=0$ and thus, even in the case $f\equiv 0$, time derivatives of $u$ will blow up as $t\to 0$. Also, in the case where $f$ is locally Lipschitz, solutions of (E) can blow up for $t>0$ in finite time. Stability and convergence of the scheme in $W^{2,1}$ is shown in the linear case without assuming $u_{tt}$ (which can blow up as $t\to 0$ is...

In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in a discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both, the perturbation parameters of the problem and the anisotropy of the mesh. The equilibrated residual method has been shown to provide one...

The tetrahedral stress element is introduced and two different types of a finite piecewise linear approximation of the dual elasticity problem are investigated on a polyhedral domain. Fot both types a priori error estimates $O\left(h^2\right)$ in $L_2$-norm and $O\left(h^{1/2}\right)$ in $L_{\infty }$-norm are established, provided the solution is smooth enough. These estimates are based on the fact that for any polyhedron there exists a strongly regular family of decomprositions into tetrahedra, which is proved in the paper, too.

The existence of a continuous right inverse of the divergence operator in $W{₀}^{1,p}\left(\Omega \right)ⁿ$, 1 < p < ∞, is a well known result which is basic in the analysis of the Stokes equations. The object of this paper is to show that the continuity also holds for some weighted norms. Our results are valid for Ω ⊂ ℝⁿ a bounded domain which is star-shaped with respect to a ball B ⊂ Ω. The continuity results are obtained by using an explicit solution of the divergence equation and the classical theory of singular integrals...

In this paper Rothe’s classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied.

In this paper, we discuss an hp-discontinuous Galerkin finite element method (hp-DGFEM) for the laser surface hardening of steel, which is a constrained optimal control problem governed by a system of differential equations, consisting of an ordinary differential equation for austenite formation and a semi-linear parabolic differential equation for temperature evolution. The space discretization of the state variable is done using an hp-DGFEM, time and control discretizations are based on a discontinuous Galerkin...

In this paper, we discuss an hp-discontinuous Galerkin finite element method (hp-DGFEM) for the laser surface hardening of steel, which is a constrained optimal control problem governed by a system of differential equations, consisting of an ordinary differential equation for austenite formation and a semi-linear parabolic differential equation for temperature evolution. The space discretization of the state variable is done using an hp-DGFEM, time and control discretizations are based on a discontinuous Galerkin...

In this paper we study the finite element approximations to the parabolic and hyperbolic integrodifferential equations and present an immediate analysis for global superconvergence for these problems, without using the Ritz projection or its modified forms.

In this article, we consider the initial value problem which is obtained after a space discretization (with space step $h$) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size $h$ chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between $h$ and the time step size...

In this article, we consider the initial value problem which is obtained after a space discretization (with space step h) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size h chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between h and the time...