a b s t r a c tThe phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for onedimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L 2 -norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.

We devise and analyze a new adaptive stabilized finite element method. We illustrate its performance on the advection-reaction model problem. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf-sup stability. We formulate this residual minimization as a stable saddle-point problem which delivers a stabilized discrete solution and an error representation that drives the adaptive mesh refinement. Numerical results on the advection-reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on uniformly refined meshes and smooth solutions. Moreover, the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers.

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In some geological formations, borehole resistivity measurements can be simulated using a sequence of 1D models. By considering a 1D layered media, we can reduce the dimensionality of the problem from 3D to 1.5D via a Hankel transform. The resulting formulation is often solved via a semi-analytic method, mainly due to its high performance. However, semi-analytic methods have important limitations such as, for example, their inability to model piecewise linear variations on the resistivity. Herein, we develop a multi-scale finite element method (FEM) to solve the secondary field formulation. This numerical scheme overcomes the limitations of semi-analytic methods while still delivering high performance. We illustrate the performance of the method with numerical synthetic examples based on two symmetric logging-while-drilling (LWD) induction devices operating at 2 MHz and 500 KHz, respectively.