A posteriori estimations of approximate solutions for some types of boundary value problems
A posteriori upper and lower error bound of the high-order discontinuous Galerkin method for the heat conduction equation
We deal with the numerical solution of the nonstationary heat conduction equation with mixed Dirichlet/Neumann boundary conditions. The backward Euler method is employed for the time discretization and the interior penalty discontinuous Galerkin method for the space discretization. Assuming shape regularity, local quasi-uniformity, and transition conditions, we derive both a posteriori upper and lower error bounds. The analysis is based on the Helmholtz decomposition, the averaging interpolation...
A Potential Method for the Biharmonic Equation.
A potential theoretic inequality
In this paper is proved a weighted inequality for Riesz potential similar to the classical one by D. Adams. Here the gain of integrability is not always algebraic, as in the classical case, but depends on the growth properties of a certain function measuring some local potential of the weight.
A potential well theory for the wave equation with a nonlinear boundary condition.
A Practical Finite Element Approximation of a Semi-Definite Neumann Problem on a Curved Domain.
A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization
In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by , where and are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.
A predator-prey model with combined death and competition terms
The existence of a positive solution for the generalized predator-prey model for two species are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.
A predictive method allowing the use of a single ionic model in numerical cardiac electrophysiology
One of the current debate about simulating the electrical activity in the heart is the following: Using a realistic anatomical setting, i.e. realistic geometries, fibres orientations, etc., is it enough to use a simplified 2-variable phenomenological model to reproduce the main characteristics of the cardiac action potential propagation, and in what sense is it sufficient? Using a combination of dimensional and asymptotic analysis, together with the well-known Mitchell − Schaeffer model, it is shown...
A Primal Hybrid Finite Element Method for Quasilinear Second Order Elliptic Problems.
A priori bounds and global existence for a strongly coupled quasilinear parabolic system modeling chemotaxis.
A priori Bounds and necessary conditions for solvability of prescribed curvature equations.
A priori bounds and renormalized Morse indices of solutions of an elliptic system
A priori bounds for a semilinear wave equation
A priori bounds for global solutions of a semilinear parabolic problem.
A priori bounds for positive radial solutions of quasilinear equations of Lane–Emden type
We consider the quasilinear equation , and present the proof of the local existence of positive radial solutions near under suitable conditions on . Moreover, we provide a priori estimates of positive radial solutions near when for is bounded near .
A priori bounds for solutions of parabolic problems and applications
A priori bounds for solutions of parabolic problems and applications
We review some recent results concerning a priori bounds for solutions of superlinear parabolic problems and their applications.
A priori convergence of the greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...
A priori convergence of the Greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...