Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged an alternative to classical for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer high approximation capabilities. However, when the PDE...

Numerical solution of multi-dimensional PDEs is a challenging problem with respect to computational cost and memory requirements, as well as regarding representation of realistic geometries and adaption to solution features. Meshfree methods such as global radial basis function approximation have been successfully applied to several types of problems. However, due to the dense linear systems th...

A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an M -matrix under some mild assumption for the underlying (generally unstructured) finite element grids. A...

In this note we show that the non-symmetric version of the classical Nitsche’s method for the weak imposition of boundary conditions is stable without penalty term. We prove optimal H1-error estimates and L2-error estimates that are suboptimal with half an order in h. Both the pure diffusion and the convection–diffusion problems are discussed.

Kolmogorov N-widths are an approximation theory concept that, for a given problem, yields information about the optimal rate of convergence attainable by any numerical method applied to that problem. We survey sharp bounds recently obtained for the N-widths of certain singularly perturbed convection-diffusion and reaction-diffusion boundary value problems.

We have defined and analyzed a semi-Toeplitz preconditioner for timedependent and steady-state convection-diffusion problems. The preconditioner exhibits very good theoretical convergence properties. The analysis is corroborated by numerical experiments.

This paper presents an a posteriori residual error estimator for diffusion– convection–reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in Rd, d = 2 or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results.

The paper presents a convergence analysis of a multigrid solver for a system of linear algebraic equations resulting from the discretization of a convection-diffusion problem using a finite element method. We consider piecewise linear finite elements in combination with a streamline diffusion stabilization. We analyze a multigrid method that is based on canonical intergrid transfer operators, a...

We derive a posteriori error estimators for convection-diffusion equations with dominant convection. The estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local meshPeclet number. The estimators are either based on the evaluation of local residuals or on the solution of discrete loc...

We present a nonlinear subgrid–scale method for the stabilization of the Galerkin approximation to convection dominated, convection diffusion problems, establish existence and uniqueness results, and provide an a priori error estimate for the method. ∗email: [email protected], Department of Mathematical Sciences, Clemson University, Clemson S.C. 29634 †email: [email protected], Department of...