1. 1 Finite Elements

2. 2 Finite elements Formally identical to spectral technique Dependent variables expanded as linear combinations of basis local, non orthogonal, linearly independent functions Orthogonalize the error (Galerkin) with respect to some test functions (no necessarily the basis functions) “Chapeau” or piecewise linear functions 1 x xjxj x j+1 x j+2 x j-1 ejej e j+1 x j : j th node ej=ej= nodal values

3. 3 Fit of a function x x j-1 xjxj x j+1 collocation minimum norm Set of coupled equations Discontinuous derivatives at the nodes

4. 4 Approximation of the derivative Coefficients of the expansion of the derivative function not derivatives of the  j ’s Galerkin: coupled system of equations R.H.S. identical to centered finite-differences scheme Accuracy can be shown to be 4 th order (superconvergence)

5. 5 Approximation of the second derivative Galerkin: integration by parts R.H.S. identical to centered finite-differences scheme

6. 6 More basis functions Quadratic elements j-1jj+1j+2j-1j-2jj+1j+2 Cubic B-splines combined with linear elements convolution polynomials j-2 j-1 j j+1 j+2 continuous f, f’ and f”

7. 7 One-dimensional linear advection equation + initial and boundary conditions 1N mass matrix Coupled system even with explicit time scheme no penalty for (stable) implicit schemes

8. 8 Boundaries can be solved as for all elements of d  /dt The solution is artificial as it doesn’t take into account the boundary conditions Fix: multiply by all e m ’s except e.g. e 1 and consider  1 not as an unknown In some cases, the matrix on the l.h.s. is singular and this indicates the need for boundary conditions e.g. calculation of f(x) from its derivative: Matrix B is singular and cann’t be inverted Sometimes the boundary conditions can be enforced with the choice of basis functions

9. 9 Irregular grids and asymmetric algorithms 0 1 x jj+1j-1j-2 basis functions test functions

10. 10 Cubic finite elements for vertical integrals at ECMWF can be approximated as Using Galerkin with test functions t k which, in matrix form is and, including the transforms to finite-element space and back so is the integration operator in matrix form

11. 11 Basis functions in the vertical at ECMWF e’s d’s top surface

12. 12 Non-linear terms Interaction coefficients Two step method Collocation Similar to the spectral case but (e j e’ k,e m ) sparse similar to the transform method is the spectral technique

13. 13 Staggered elements In the gravity-wave equations we can expand u(x) in terms of the e j ’s expand h(x) in terms of the e j+1/2 ’s use the e j ’s as test functions in the u-equation use the e j+1/2 ’s as test functions in the h-equation j-1j+1j j-1/2j+1/2 j+3/2 e j ’s e j+1/2 ’s

14. 14 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 2-D elements x x x x x x x e i (x).e j (y) hexagonal elements irregular elements used in engineering

15. 15 The local spectral technique Within every finite element, the function is represented not by a single polynomial but by a combination of some basis functions, such as Chebishev polynomials Lagrange interpolating polynomials Multigrid-like methods applicable Advantage of the spectral method inside each element Advantages of the finite-element method for complex geometries