Exact Bounds for the Solution Branches of Nonlinear Eigenvalue Problems.
Exact solutions for the generalized BBM equation with variable coefficients.
Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows
In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are established using a Newton-chord fixed point theorem, a posteriori error estimates are also derived....
Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows
In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are established using a Newton-chord fixed point theorem, a posteriori error estimates are also derived. An...
Existence and Approximation of Solutions of Non-Linear Elliptic Equations.
Existence and Approximation Results for Nonlinear Mixed Problems : Application to Incompressible Finite Elasticity.
Existence de maillages optimaux dans les méthodes d'éléments finis
Existence of nontrivial solution for a nonlocal elliptic equation with nonlinear boundary condition.
Existence of solutions to singular elliptic equations with convection terms via the Galerkin method.
Existenzsätze mit der Linienmethode für Parabolische Probleme und periodische Lösungen von ut = f(t,x,u,ux,uxx).
Expanded mixed finite element methods for linear second-order elliptic problems, I
Expanded mixed finite element methods for quasilinear second order elliptic problems, II
Explicit Difference Approximations of Du Fort-Frankel Type of the One-Dimensional Diffusion Equation.
Explicit estimation of error constants appearing in non-conforming linear triangular finite element method
The non-conforming linear () triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both the theoretical and practical purposes. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuška-Aziz maximum angle condition is required just as in the case of the conforming triangle. Some applications...
Explicit finite element error estimates for nonhomogeneous Neumann problems
The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle over finite element spaces is constructed and the explicit upper bound of the constant in the trace theorem is given. Numerical examples are shown in the final section, which implies the proposed error estimate has the convergence rate as .
Explicit Finite Element Schemes for First Order Symmetric Hyperbolic Systems.
Exponential convergence of hp quadrature for integral operators with Gevrey kernels
Galerkin discretizations of integral equations in require the evaluation of integrals where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for xy and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules using N function evaluations of g which achieves exponential convergence |I – | ≤C exp(–rNγ) with...
Exponential convergence of hp quadrature for integral operators with Gevrey kernels
Galerkin discretizations of integral equations in require the evaluation of integrals where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for xy and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules using N function evaluations of g which achieves exponential convergence |I – | ≤C exp(–rNγ) with...
Exponential expressivity of neural networks on Gevrey classes with point singularities
We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains , . We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in , comprising the countably-normed spaces of I. M. Babuška and B. Q. Guo. As intermediate result, we prove that continuous, piecewise polynomial high...
Extending Babuška-Aziz's theorem to higher-order Lagrange interpolation
We consider the error analysis of Lagrange interpolation on triangles and tetrahedrons. For Lagrange interpolation of order one, Babuška and Aziz showed that squeezing a right isosceles triangle perpendicularly does not deteriorate the optimal approximation order. We extend their technique and result to higher-order Lagrange interpolation on both triangles and tetrahedrons. To this end, we make use of difference quotients of functions with two or three variables. Then, the error estimates on squeezed...