Adaptive ACMS: A robust localized Approximated Component Mode Synthesis Method

We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follows the Variational Multiscale and Localized Orthogonal Decomposition--LOD approaches in the sense that it decouples spaces into multiscale and fine subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, however it deteriorates with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space $H^1$

Song patterns: elementary book for violin

Thesis (M.A.)--Boston UniversityThe purpose of this thesis is to present certain concepts and the necessary materials which will lead toward the development of a thorough foundation for the beginner. This purpose is not new, for it is the intent of any violin method to develop the playing ability of the student, but the approach used is unique in the opinion of the author. An introduction is made by the use of open string double-stops. With the use of double stops, the pupil has a double means of control. He must not only hear two pitches sounding simultaneously, but must also see that his bow is in contact with two strings at all times. He must maintain a steady movement of the bow arm on the same plane from the frog to the tip of the bow. The resulting bow movement should be free, firm and steady. The introduction of spiccato and staccato to a pupil at this beginning period, is made to bring about a feeling of familiarity with the instrument and a certain degree of control of the bow. From this point on the emphasis is placed on ear training. The correct placement of the third finger of the left hand is determined by adjustment until a perfect octave is formed with the adjacent lower string. With pupils whose hands are fairly well developed in size, the next step will be the placement of the fourth finger in correct position to form a perfect unison with the adjacent higher string. After the correct position is found for the third finger, the second finger is placed on the string close to the third thus forming Fa and Mi. Re and Do are introduced by using the first finger and the open string in that order. The formation of the tetrachord by placing fingers 3, 2, 1 and the open string gives us the basis of the entire text. The tetrachord is given in five positions so that the pupil becomes familiar with the key of C, G, D, A, E, B, F, Bb, Eb and Ab. Nevertheless, the tonal pattern of the tetrachord and the relationship within the four tones of the group remains the same in every key. The pupil has experienced in his schoolroom vocal study the pattern Fa-Mi-Re-Do. He is able to recognize it and when necessary sing it. He is able to determine the placement of each finger on the fingerboard by remembering that only those fingers used to play the third and fourth steps of the tetrachord will be placed in a close position. There is no mention made of high or low finger positions. The placement of each finger is determined aurally. The rhythmic development parallels the melodic. As each new position of the tetrachord is mastered a new rhythmic problem is introduced. The pupil becomes familiar with quarter, half, dotted-half, whole, eighth, dotted quarter, sixteenth notes, their corresponding rests and six-eight time, in the order mentioned. The author has given no bowing indications for a very definite reason. He has felt that an introduction to bowing and slurring can be made after the problems in this method have been mastered. He feels that there will be no great handicap if this introduction is postponed until that time. He has intended to stress good intonation and its development as the primary objective. With the exception of "America" every song used is a folksong, The songs used are all very melodic, a majority of them familiar and all of a type within a young pupils range of appeal and interest. They have been placed in the text, and in a few instances altered for use with a particular problem. The find feature is also instrumental in making this text unusual. Each song is repeated in many ways. It is presented on different strings so that it is played in a new key each time, yet the finger pattern of the tetrachord remains the same on each string. The songs are also used for review purposes. After a song is mastered in the new pattern, it is presented in the key of one of the pattern previously learned, thus providing a bridge between the new and the old

Hybrid Localized Spectral Decomposition for multiscale problems

We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the unknowns through elliptic problems and satisfies equilibrium constraints. One of the resulting problems is non-local but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. We obtain quasi-optimal a priori error estimates for low-contrast problems assuming minimal regularity of the solutions. To also consider the high-contrast case, we propose a variant of our method, enriching the space solution via local eigenvalue problems and obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes and again assuming no regularity of the solution. The technique developed is dimensional independent and easy to extend to other problems such as elasticity