• S.L. Manikonda
  ANL, Argonne
• M. Berz, K. Makino
  MSU, East Lansing, Michigan

Many processes in Physics can be described by Partial Differential equations (PDE’s). For various practical problems, very precise and verified solutions of PDE are required; but with conventional finite element or finite difference codes this is difficult to achieve because of the need for an exceedingly fine mesh which leads to often prohibitive CPU time. We present an alternative approach based on high-order quadrature and a high-order finite element method. Both of the ingredients become possible through the use of Differential Algebra techniques. Further the method can be extended to provide rigorous error verification by using the Taylor model techniques. Application of these techniques and the precision that can be achieved will be presented for the case of 3D Laplace’s equation. Using only around 100 finite elements of order 7, verified accuracies in the range of 10^-7 can be obtained.