r-refinement for evolutionary PDEs with finite elements or finite differences

Positive Finite Elements for Hyperboli PDEs

Finite Elements with mesh refinement for wave equations in polygons

Error estimates for the space-semidiscrete approximation of solutions of the wave equation in polygons G ⊂ R are presented. Based on corner asymptotics of solutions of the wave equation, it is shown that for continuous, simplicial Lagrangian Finite Elements of polynomial degree p ≥ 1 with either suitably graded mesh refinement or with bisection tree mesh refinement towards the corners of G, the...

Mixed Finite Elements for Elliptic Problems with Tensor Coeecients as Cell-centered Finite Diierences Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-centered Finite Differences

We present an expanded mixed nite element approximation of second order elliptic problems containing a tensor coeecient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its ux (the tensor coeecient times the negative gradient). The resulting linear system is a saddle point problem. In the c...

Mesh-Centered Finite Differences from Nodal Finite Elements

After it is shown that the classical ve points mesh-centered nite diierence scheme can be derived from a low order nodal nite element scheme by using nonstandard quadrature formulae, higher order block mesh-centered nite diierence schemes for second-order elliptic problems are derived from higher order nodal nite elements with nonstandard quadrature formulae as before, combined to a procedure k...

Finite differences and finite elements: getting to know you

T o debug your programs, it’s helpful to experiment with the simplest test problem and a small number of mesh points. Look ahead to Problem 6 for sample problems. Problem 2 uses the Matlab function spdiags to construct a sparse matrix. If you have never used sparse matrices in Matlab, print the matrix A to see that its data structure contains the row index, column index, and value for each nonz...