A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods

We develop computable a posteriori error estimates for the pointwise evaluation of linear functionals of a solution to a parameterized linear system of equations. These error estimates are based on a variational analysis applied to polynomial spectral methods for forward and adjoint problems. We also use this error estimate to define an improved linear functional and we prove that this improved functional converges at a much faster rate than the original linear functional given a pointwise convergence assumption on the forward and adjoint solutions. The advantage of this method is that we are able to use low order spectral representations for the forward and adjoint systems to cheaply produce linear functionals with the accuracy of a higher order spectral representation. The method presented in this paper also applies to the case where only the convergence of the spectral approximation to the adjoint solution is guaranteed. We present numerical examples showing that the error in this improved functional is often orders of magnitude smaller. We also demonstrate that in higher dimensions, the computational cost required to achieve a given accuracy is much lower using the improved linear functional.

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