Numerical approximation of Poisson problems using high-order continuous Galerkin methods with static condensation
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Higher-order methods in finite elements can provide better approximations than linear methods, in some problems. This is because they can offer an exponential convergence rate of the solution. Thus, in some applications, high-order methods can be cheaper than low-order methods. Nonetheless, it is of major importance to provide good implementations in order to reduce the computational cost of solving a problem. To this end, we propose to use the classical continuous Galerkin method with static condensation procedure to reduce the memory footprint and the CPU time. The main idea consists on write the unknowns related to the inner nodes of each element in terms of the unknowns related to the boundary nodes of the elements. Thus, this method effectively suppress all the unknowns that correspond to pure interior elemental nodes. To show these properties, we apply the static condensation technique to the Poisson problem. We will particularize the proposed technique for this problem, and we will compare the obtained solution and the computational cost with a classical implementation of the high-order continuous Galerkin method. In order to formulate the method correctly, all the needed results are introduced. The results show that static condensation is a valid choice since it reduces the computational cost of solving a problem.